CUET Mathematics Algebra > Matrices & DeterminantsFree, no login required.

For any non-zero matrix $A = [a_{ij}]_{3\times3}$ and $|A| \neq 0$, then Match the LIST-I with LIST-II | | LIST-I | | LIST-II | |---|---|---|---| | A. | $\vert adj\,A \vert$ | I. | $\dfrac{1}{\vert A \vert}(adj\,A)$ | | B. | $(adj\,A)A$ | II. | $\vert A \vert^2$ | | C. | $A^{-1}$ | III. | $\dfrac{1}{\vert A \vert}$ | | D. | $\vert A^{-1} \vert$ | IV. | $\vert A \vert I$ | Choose the correct answer from the options given below:

If $A$ and $B$ are invertible matrices of order 3. If $|A| = 2$ and $|(AB)^{-1}| = \dfrac{-1}{6}$, then the value of $|B|$ is :

If $A = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -2 \\ 3 & -4 \\ 2 & 4 \end{bmatrix}$, then $AB$ is :

Let $A = [a_{ij}]_{3\times3}$ be a matrix such that $a_{ij} = \dfrac{i-j}{i+j}$, then which of the following statements is/are TRUE? A. A is a symmetric matrix B. A is a skew symmetric matrix C. $|A| = 0$ D. $|A| \neq 0$ Choose the correct answer from the options given below:

If $\begin{vmatrix} 1 & -2 & 5 \\ 2 & \lambda & -1 \\ 0 & 4 & 2\lambda \end{vmatrix} = 86$, the sum of the values of $\lambda$ is:

The number of all possible matrices of order $2 \times 2$ with each entry 0 or 1 having their determinants as positive value is :

If $A = \begin{bmatrix} 2 & -3 & -5 \\ -1 & 4 & 5 \\ 1 & -3 & -4 \end{bmatrix}$, then $A^2$ is equal to :

If $\begin{bmatrix} 0 & x \\ 2 & 1 \end{bmatrix} + 2\begin{bmatrix} 3 & 1 \\ y & 0 \end{bmatrix} = \begin{bmatrix} 6 & 5 \\ 8 & 1 \end{bmatrix}$, then the value of $2x+3y$ is :

If $a, b$ and $c$ are non-zero real number such that $\begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix} = 0$, then value of $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$ is

Let $A = \begin{bmatrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ \lambda & 2 & -3 \end{bmatrix}$. If $A^{-1}$ does not exist, then $\lambda =$

Let $A = \begin{bmatrix} 2 & 4 \\ -2 & 8 \end{bmatrix}$. If $A^{-1} = \alpha I + \beta A$, $\alpha,\beta \in R$ (Where R is set of real numbers) and I is a $2\times2$ identity matrix, than the value of $4(\alpha-2\beta)$ is equal to:

Match the LIST-I with LIST-II | | LIST-I<br>Matrix A | | LIST-II<br>$\vert adj\,A \vert$ | |---|---|---|---| | A. | $\begin{bmatrix}1 & 2 & 3\\ 0 & 2 & 5\\ 0 & 0 & 1\end{bmatrix}$ | I. | $9$ | | B. | $\begin{bmatrix}3 & -1 & 4\\ 0 & 1 & 2\\ 0 & 0 & -1\end{bmatrix}$ | II. | $16$ | | C. | $\begin{bmatrix}4 & 2 & -3\\ 0 & -2 & 5\\ 0 & 0 & 1\end{bmatrix}$ | III. | $64$ | | D. | $\begin{bmatrix}2 & 5 & 3\\ 0 & 2 & 1\\ 0 & 0 & 1\end{bmatrix}$ | IV. | $4$ | Choose the correct answer from the options given below:

If A is a square matrix of order $n$ and $k$ is a scalar such that $|kA| = \alpha|A|$ then the value of $\alpha$ is:

If the matrix $\begin{bmatrix} 5 & x^2 \\ 3x-2 & x-1 \end{bmatrix}$ is symmetric, then values of $x$ are :

If matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$, then $A^2+I$ is equal to: (Where I is an identity matrix of order 2)

If $a_{ij}$ and $A_{ij}$ represents the elements and cofactors of the matrix $A = \begin{bmatrix}2 & 1 & 3\\3 & 2 & 1\\1 & 2 & 3\end{bmatrix}$, then which of the following statements are True? (A) $|A| = 28$ (B) $A_{23} = -3$ (C) $a_{11}A_{12} = -16$ (D) $\frac{A_{21}}{A_{13}} = \frac{3}{4}$ Choose the correct answer from the options given below:

If $A = \begin{bmatrix}0 & a & 1\\-1 & b & 1\\-1 & c & 0\end{bmatrix}$ is a skew symmetric matrix, then the value of $(a+b-c)^2$ is

A is a square matrix of order 3 x 3, then Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) $\left\vert \text{adj}A \right\vert$ | (I) $\dfrac{1}{\left\vert A \right\vert}$ | | (B) $\left\vert A^{-1} \right\vert$ | (II) $\left\vert A \right\vert^{4}$ | | (C) $\left\vert A.\text{adj}A \right\vert$ | (III) $\left\vert A \right\vert^{3}$ | | (D) $\left\vert \text{adj}(\text{adj}A) \right\vert$ | (IV) $\left\vert A \right\vert^{2}$ | Choose the correct answer from the options given below:

If A and B are two square matrices of the same order such that $(A+B)^2 = A^2+B^2$, then

The value of $\begin{vmatrix}x & a & a\\a & x & a\\a & a & x\end{vmatrix}$ is equal to:

If $A = \begin{bmatrix}\cos\alpha & \sin\alpha\\-\sin\alpha & \cos\alpha\end{bmatrix}$, then the value of $1000|A^n| + 2|A|$ for a positive integer $n$, is equal to

If A is square matrix of order 3 x 3 such that $a_{ij} = j - i$, then which of the following statements are correct? (A) A is a symmetric matrix. (B) A is a skew symmetric matrix. (C) $|A| = 0$. (D) A is a diagonal matrix. Choose the correct answer from the options given below:

If A and B are invertible matrices of the same order, then Match **List-I** with **List-II** | List-I (Determinant) | List-II (Equal to) | |---|---| | (A) $\left\vert A^{-1}B \right\vert$ | (I) $\dfrac{1}{\left\vert A \right\vert \left\vert B \right\vert}$ | | (B) $\left\vert AB^{-1} \right\vert$ | (II) $\dfrac{\left\vert B \right\vert}{\left\vert A \right\vert}$ | | (C) $\left\vert A^{-1}B^{-1} \right\vert$ | (III) $\left\vert A \right\vert \left\vert B \right\vert$ | | (D) $\left\vert AB \right\vert$ | (IV) $\dfrac{\left\vert A \right\vert}{\left\vert B \right\vert}$ | Choose the correct answer from the options given below:

If $A = \begin{bmatrix}1 & 2\\2 & 1\end{bmatrix}$ and $f(x) = x^2-2x-3$, then $f(A)$ is

If $A = \begin{bmatrix}\frac{1}{4} & 2\\0 & 2x-3\end{bmatrix}$, $B = \begin{bmatrix}4 & 8\\0 & -1\end{bmatrix}$ and AB = $I_2$ (Identity matrix of order 2), then the value of $x$ is

If $A = \begin{bmatrix}\alpha & \beta\\r & \Delta\end{bmatrix}$, such that $|A| = 25$, then the value of $\left|\frac{1}{5}A\right|$ is.

If $A = \begin{bmatrix}0 & x & 3\\2 & y & -1\\z & 1 & 0\end{bmatrix}$ is a skew-symmetric matrix, then the value of $3x+y-2z$ is

If A and B are symmetric matrices of same order, then which of the following statements are true? (A) $AB^T+B^TA$ is a skew-symmetric matrix. (B) $AB^T-BA^T$ is a skew-symmetric matrix (C) $(AB)^T(A^T)^{-1}=B$ (D) $(AB)^{-1}(BA)^T=A$ Choose the correct answer from the options given below:

Let A be a square matrix of order 3 such that $|A|=3$, then the value of $\frac{|adj(adjA)|}{|A|}$ is

If A and B are square matrices of order n, such that AB = A and BA = B, then $A^2$ =

The value (/s) of $x$ satisfying the matrix equation: $x\begin{bmatrix}2x & 2\\3 & x\end{bmatrix}+2\begin{bmatrix}8 & 5x\\4 & 4x\end{bmatrix}=2\begin{bmatrix}(x^2+8) & 24\\10 & 6x\end{bmatrix}$ is/are:

Let A, B be square matrices of order $3\times3$. Then $|A.adj(A).B|$ is

If A and B are two symmetric matrices of the same order, then

Match the LIST-I with LIST-II | | LIST-I<br>(Matrix) | | LIST-II<br>(Type of matrix) | |---|---|---|---| | A. | $A = [a_{ij}]_{m \times m}$ where $a_{ij} = 0\ \forall\, i \neq j$ | I. | Scalar matrix | | B. | $A = [a_{ij}]_{m \times 1}$ | II. | Diagonal matrix | | C. | $A = [a_{ij}]_{1 \times n}$ | III. | Column matrix | | D. | $A = [a_{ij}]_{m \times m}$ where $a_{ij} = 0\ \forall\, i \neq j$ and $a_{ij} = k\ \forall\, i = j, k \neq 0$ | IV. | Row matrix | Choose the correct answer from the options given below:

Let $A=\begin{bmatrix}1 & \sin\theta & 1\\-\sin\theta & 1 & \sin\theta\\-1 & -\sin\theta & 1\end{bmatrix}$, where $\theta\in[0,2\pi]$ then which of the following statements are correct? A. $|A|=2+2\sin^2\theta$ B. Maximum value of $|A|$ is 1. C. Maximum value of $|A|$ is –1. D. $|A|\in[2,4]$ Choose the correct answer from the options given below:

The value of the determinant $\Delta=\begin{vmatrix}\cos\alpha\cos\beta & \cos\alpha\sin\beta & -\sin\alpha\\-\sin\beta & \cos\beta & 0\\\sin\alpha\cos\beta & \sin\alpha\sin\beta & \cos\alpha\end{vmatrix}$ is:

If $A=\begin{bmatrix}2 & 3\\1 & 2\end{bmatrix}, I=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$ and $A^2=\alpha A+\beta I$, for some constant $\alpha$ and $\beta$ then the value of $\alpha$ and $\beta$ respectively:

Match the LIST-I with LIST-II | | LIST-I<br>Number of the elements in a matrix | | LIST-II<br>Number of matrices of possible order | |---|---|---|---| | A. | $17$ | I. | $8$ | | B. | $8$ | II. | $2$ | | C. | $12$ | III. | $6$ | | D. | $24$ | IV. | $4$ | Choose the correct answer from the options given below:

The value of $\begin{vmatrix}1 & \log_ba\\\log_ab & 1\end{vmatrix}$ is:

If $A=\begin{bmatrix}1 & 2\\2 & 1\end{bmatrix}$ and $f(x)=x^2-2x-3$, then $f(A)$ is equal to

If $A$ is a square matrix of order 3 and $|A|=6$, then the value of $|2A^T|$ is:

If $A$ is a square matrix of order 3 such that $A(adj\,A)=\begin{bmatrix}-3 & 0 & 0\\0 & -3 & 0\\0 & 0 & -3\end{bmatrix}$, then which of the following is NOT correct?

If the matrix $\begin{bmatrix}0 & \alpha & 2\\2 & \beta & 1\\\gamma & -1 & 0\end{bmatrix}$ is skew-symmetric, then $(\beta-\alpha-\gamma)$ equals

Which of the following statements are correct? A. If $P$ and $Q$ are square matrices of same order, then $(PQ^T-QP^T)$ is a skew symmetric matrix. B. The number of all possible matrices of order $3\times2$ with each entry either –1 or 1, is 64. C. If $P$ is an invertible matrix, then $P^T$ is also invertible. D. If $P$ and $Q$ are square matrices of the same order, then $|PQ|=|P|+|Q|$. Choose the correct answer from the options given below:

For what values of $\lambda$ and $\mu$, the following system of linear equations: $x+y+2z=1, 2x+y-z=4, x+2y+\lambda z=\mu$ have infinitely many solution?

If the matrix $A = \begin{bmatrix} -2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{bmatrix}$ is given, then which of the following statements is/are TRUE? A. A is a square matrix B. A is a symmetric matrix C. A is a scalar matrix D. A is a identity matrix Choose the correct answer from the options given below:

If $A+B = \begin{bmatrix}7 & 0\\2 & 5\end{bmatrix}$ and $A-B = \begin{bmatrix}3 & 0\\0 & 3\end{bmatrix}$, then

If A is a diagonal matrix of order 3 with diagonal elements as 2, 3, 4, then |adj A| is

If area of the triangle formed by A(1, 3), B(0, 0) and C(k, 0) is 3 sq. units, then which of the following is/are TRUE? A. k = 2 B. k = -2 C. k = 3 D. k = -3 Choose the correct answer from the options given below:

If A and B are square matrices of order 3×3 such that |A|=7 and |B|=6, then |B adj A| =

If $A=\begin{bmatrix}1&1&1\\1&0&2\\3&1&1\end{bmatrix}$ and $B=\begin{bmatrix}-2&0&2\\5&-2&-1\\1&2&-1\end{bmatrix}$ such that AB=4I (where I is identity matrix) then solution of system of equations $x+y+z=6, x+2z=7, 3x+y+z=12$ is

If $\begin{vmatrix}2a & 2a & 3b+c\\3b & 2a+c & 3b\\2a+3b & c & c\end{vmatrix} = 48$, then abc is equal to

If $\begin{vmatrix}x&2\\18&x\end{vmatrix} = \begin{vmatrix}6&2\\18&6\end{vmatrix}$, then x equals:

Let $A=[a_{ij}]_{3\times3}$, where $a_{ij}=i-j$, then which of the following statements is/are TRUE ? A. A is diagonal matrix B. A is square matrix C. A is symmetric matrix D. A is skew symmetric matrix Choose the correct answer from the options given below:

If A is a skew symmetric matrix, then $A^2$ is a

The determinant of which of the following matrices is a perfect square ? A. $\begin{bmatrix}5&29\\-1&6\end{bmatrix}$ B. $\begin{bmatrix}5&26\\1&6\end{bmatrix}$ C. $\begin{bmatrix}10&3\\17&10\end{bmatrix}$ D. $\begin{bmatrix}8&5\\-3&-8\end{bmatrix}$ Choose the correct answer from the options given below:

With reference to the matrix $A=\begin{bmatrix}5&0&0\\0&5&0\\0&0&-5\end{bmatrix}$, which of the following are TRUE ? A. |A|=125 B. A is a scalar matrix. C. A is a diagonal matrix. D. |A|=-125 Choose the correct answer from the options given below:

Match the LIST-I with LIST-II | LIST-I | LIST-II | |---|---| | A. $\begin{vmatrix} x+2 & 3 \\ x+5 & 4 \end{vmatrix} = 3$ | I. $x = 10$ | | B. $\begin{vmatrix} x+2 & x+5 \\ 3 & 4 \end{vmatrix} = 13$ | II. $x = 60$ | | C. $\begin{vmatrix} x+4 & x-4 \\ 8 & 7 \end{vmatrix} = 0$ | III. $x = 0$ | | D. $\begin{vmatrix} x+4 & 8 \\ x-4 & 7 \end{vmatrix} = 60$ | IV. $x = 20$ | Choose the correct answer from the options given below:

If matrix $A=\begin{bmatrix}0&3a&-4\\2&1&2\\2b&2&-1\end{bmatrix}$ is a symmetric matrix, then the values of $a$ and $b$ are

Given that $[a_{ij}]$ is a square matrix of order n. Match the LIST-I with LIST-II | LIST-I | | LIST-II | |---|---|---| | A. | $A=[a_{ij}]; a_{ij}=\begin{cases}1, & \forall i=j\\0, & \forall i\neq j\end{cases}$ | I. Identity Matrix | | B. | $B=[a_{ij}]; a_{ij}=\begin{cases}2i-3j, & \forall i=j\\0, & \forall i\neq j\end{cases}$ | II. Diagonal Matrix | | C. | $C=[a_{ij}]; a_{ij}=\begin{cases}k, & \forall i=j\\0, & \forall i\neq j\end{cases}$ | III. Scalar Matrix | | D. | $D=[a_{ij}]; a_{ij}=\begin{cases}2i-3j, & \forall i\leq j\\0, & \forall i>j\end{cases}$ | IV. Upper Triangular Matrix | Choose the correct answer from the options given below:

Let $A = \begin{bmatrix} 1 & -3 \\ 1 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 4 & 6 \\ -2 & 2 \end{bmatrix}$ and $C$ be a matrix such that $C = AB$. Then $|(BC)^{-1}|$ is equal to

If $A = \begin{bmatrix} 2 & 1 & -1 \\ 0 & 3 & -2 \\ k & 1 & 0 \end{bmatrix}$ is non-singular matrix, then the value(s) of $k$ belongs to: [Where $R$ is the set of real numbers]

Let $A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}$ and $I$ be an identity matrix of order 3. If $A^TA=I$, then $x^2+y^2+z^2$ is equal to:

Let $A=[a_{ij}]$ be a matrix of order $3 \times 4$ such that $a_{ij}=\begin{cases}2i-3j, & i \ge j \\ 3i+j, & i<j\end{cases}$ then the sum of all the elements of the first two rows of this matrix is:

If $A=\begin{bmatrix} x & p & x+p \\ y & q & y+q \\ z & r & z+r \end{bmatrix}$, then $|A|$ is

If $\begin{bmatrix} x-2 & 3 & y \\ -3 & 0 & 2 \\ 4 & z-1 & 0 \end{bmatrix}$ is a skew-symmetric matrix, then the value of $x+y+z$ is:

Let $A=\begin{bmatrix}1 & 0 \\ -1 & 7\end{bmatrix}$ and $B=\begin{bmatrix}0 & 4 \\ -1 & 7\end{bmatrix}$ then Match List-I with List-II | List-I | List-II | |---|---| | **Determinants** | **Value** | | (A) $\left\vert A^{2} \right\vert$ | (I) $28$ | | (B) $\left\vert AB^{T} \right\vert$ | (II) $7$ | | (C) $\left\vert \text{adj } A \right\vert$ | (III) $16$ | | (D) $\left\vert 2B \right\vert$ | (IV) $49$ | Choose the correct answer from the options given below:

$\begin{vmatrix} a & b-c & c+b \\ a+c & b & c-a \\ a-b & a+b & c \end{vmatrix}$ is equal to:

Let $A=\begin{bmatrix}2 & 2 & 2 \\ 0 & 2 & 2 \\ 0 & 0 & 2\end{bmatrix}$, then $A^6$ is equal to:

Let $A=\begin{bmatrix}a & 0 \\ b & 0\end{bmatrix}$ and $A^2=\begin{bmatrix}4 & c \\ 2 & 0\end{bmatrix}$, $a,b,c \in \mathbb{Z}$. Then the minimum value of $a+b+c$ is:

If A is a non-singular matrix of order 3, then Match List-I with List-II | List-I | List-II | |---|---| | Determinant of A | $\left\vert \text{adj}(\text{adj } A) \right\vert =$ | | (A) $\left\vert A \right\vert = 3$ | (I) $1$ | | (B) $\left\vert A \right\vert = 4$ | (II) $16$ | | (C) $\left\vert A \right\vert = 2$ | (III) $81$ | | (D) $\left\vert A \right\vert = 1$ | (IV) $256$ | Choose the correct answer from the options given below:

If $A=\begin{bmatrix}3 & -2 & 3 \\ 2 & 1 & -1 \\ 4 & -3 & 2\end{bmatrix}$ and $A(adjA)=xI$, where $I$ is an identify matrix of order 3, then $x$ is equal to

Match List-I with List-II | List-I (Singular matrix) | List-II (A value of $x$) | |---|---| | (A) $A=\begin{bmatrix}x & 4 \\ 2 & 2x\end{bmatrix}$ | (I) 3 | | (B) $A=\begin{bmatrix}3 & x \\ 3x & 9\end{bmatrix}$ | (II) 4 | | (C) $A=\begin{bmatrix}x & 1 \\ 1 & x\end{bmatrix}$ | (III) 2 | | (D) $A=\begin{bmatrix}x & 1 \\ 16 & x\end{bmatrix}$ | (IV) 1 | Choose the correct answer from the options given below:

If $A=\begin{bmatrix}2 & -1 & 3 \\ -3 & 4 & 5\end{bmatrix}$ and $B=\begin{bmatrix}2 & 1 \\ 3 & 5 \\ 2 & 3\end{bmatrix}$ and $BA=(b_{ij})$, then $b_{22}+b_{31}$ is :

Which of the following are correct? (A) If $A=\begin{bmatrix}7 & 5 \\ 4 & 3\end{bmatrix}$, then $A^{-1}=\begin{bmatrix}3 & -5 \\ -4 & 7\end{bmatrix}$ (B) If $A=\begin{bmatrix}3 & 5 \\ 6 & 10\end{bmatrix}$, then $A$ is a non-singular matrix (C) $A=\begin{bmatrix}0 & -5 \\ 5 & 3\end{bmatrix}$ is a symmetric matrix (D) If $A=\begin{bmatrix}6 & x \\ y & 0\end{bmatrix}$ and $A=A^T$, then $x=y$ Choose the correct answer from the options given below:

The real value of $a$ for which the system of equations $x+ay=0, y+az=0, z+ax=0$ has infinitely many solutions, is :

If A and B are symmetric matrices of the same order then which of the following statements are correct? A. AB is symmetric matrix B. AB - BA is skew symmetric matrix C. $B^TAB$ is symmetric matrix D. $A - A^T$ is skew symmetric matrix Choose the correct answer from the options given below:

If $A = \dfrac{1}{9}\begin{bmatrix} -8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & -8 & 4 \end{bmatrix}$, then (Where $A^{-1}$ and $A^T$ represent inverse and transpose of matrix $A$ respectively)

If a matrix $A = \begin{bmatrix} x+3 & 3 \\ 2x+1 & 5 \end{bmatrix}$ is singular , then the value of $x$ will be

If $[2x \quad 4]\begin{bmatrix} x \\ -8 \end{bmatrix} = [0]$, then the value of $x$ is/are A. $-4$ B. $4$ C. $16$ D. $-16$ Choose the correct answer from the options given below:

If A is a square matrix of order 4 and $|adj\,A| = 8$, then $|A| =$

Which of the following statements are correct ? A. If the product of 2 non-zero square matrices is a zero matrix, then either A is singular or B is singular B. If A is a non singular matrix , then $|A^{-1}| = \dfrac{1}{|A|}$ C. If A is an invertible square matrix then $adj\,A^T = \dfrac{1}{adj\,A}$ D. If A and B are invertible matrices then $(AB)^{-1} = B^{-1}A^{-1}$ Choose the correct answer from the options given below:

The value/s of '$\lambda$' for which the system of equations $\lambda x + 3y - z = 1$ $x + 2y + z = 2$ $-\lambda x + y + 2z = -1$ has unique solution is/are: (where $R$ is set of real numbers)

The matrix $\begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}$ is

If $\begin{vmatrix} 1 & x & x^2 \\ x^2 & 1 & x \\ x & x^2 & 1 \end{vmatrix} = (x^{\lambda}-1)^{\mu}$ then the value of $\lambda$ and $\mu$ are given by

If A is a square matrix of order 3 such that $|A| = 3$, then the value of $|adj\,A| + |adj\,(adj\,A)|$ is

If $A = \begin{bmatrix} 9 & 7 & 5 \\ 6 & 4 & 2 \\ 1 & 3 & 8 \end{bmatrix}$, then (cofactor of $a_{13}$ - cofactor of $a_{32}$ )will be equal to :

Match the LIST-I with LIST-II | | LIST-I | | LIST-II | |---|---|---|---| | A. | Column matrix | I. | $\begin{bmatrix}1 & 2 & 3\end{bmatrix}$ | | B. | Row matrix | II. | $\begin{bmatrix}1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3\end{bmatrix}$ | | C. | Diagonal matrix | III. | $\begin{bmatrix}1\\ 2\\ 3\end{bmatrix}$ | | D. | Scalar matrix | IV. | $\begin{bmatrix}2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\end{bmatrix}$ | Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 1 & 0 \\ -1 & 7 \end{bmatrix}$ is a matrix such that $A^2 = 8A + KI$,where $I$ is an identity matrix of order 2 then value of $K$ is :

If A is a skew-symmetric matrix then, the matrix $B^TAB$ is:-

Match the LIST-I with LIST-II | LIST-I | LIST-II | |---|---| | A. If A is an invertible square matrix then $\lvert A^{-1}\rvert =$ | I. $\dfrac{1}{\lvert A\rvert}(\mathrm{adj}\,A)$ | | B. For every square matrix (adj A) A = | II. $\dfrac{1}{\lvert A\rvert}$ | | C. For a square matrix A, if $\lvert A\rvert \neq 0$, then $A^{-1} =$ | III. $B^{-1}A^{-1}$ | | D. If A and B are invertible matrices, then $(AB)^{-1} =$ | IV. $\lvert A\rvert.I$, where I is the identity matrix of same order as A | Choose the correct answer from the options given below:

For a square matrix $A$ of order 3, if $\lvert A\rvert = -3$, then $\lvert 4\,adj A\rvert$ is equal to

Which of the following statements are TRUE? A. If the product of two matrices is a zero matrix, it is not necessary that one of the matrix is a zero matrix. B. For any three matrices A, B and C, $(AB)C = A(CB)$. C. Diagonal elements of a skew symmetric matrix can be non-zero. D. Multiplication of diagonal matrices of same order is always commutative. Choose the correct answer from the options given below:

If $P = \dfrac{1}{14}\begin{bmatrix} 3 & 2 \\ -4 & 2 \end{bmatrix}$ and $P^{-1} = \begin{bmatrix} x & -2 \\ 4 & y \end{bmatrix}$, then the value of $x$ and $y$ respectively are

If A is a $2 \times 3$ matrix and B is a matrix such that $A^{T}B$ and $BA^{T}$ are both defined. Then order of B is

The value of $\Delta = \begin{vmatrix} 2\sin40^\circ & 2\sin50^\circ \\ -3\cos40^\circ & 3\cos50^\circ \end{vmatrix}$ is:

If $\begin{vmatrix} 2x & 2 \\ 8 & x \end{vmatrix} = \begin{vmatrix} -4 & -2 \\ 4 & -2 \end{vmatrix}$ then $x =$

If A and B are two non-singular matrices of same order, then which of the following statements are not correct ? A. AB is non singular. B. $(A+B)^{-1} = B^{-1}+A^{-1}$. C. adj (AB) = adj B. adj A. D. AB is not invertible. Choose the correct answer from the options given below:

If the sum of a matrix A and its transpose is $\begin{bmatrix} -6 & 4 \\ 4 & 10 \end{bmatrix}$, then A is:

For a square matrix $A = \begin{bmatrix} 2 & 1 & 3 \\ 0 & -1 & 4 \\ 1 & 2 & 0 \end{bmatrix}$, which of the following statements is/are TRUE ? A. det (A) = 9. B. $A_{21}.A_{33} = -12$, where $A_{ij}$ is the cofactor of $a_{ij}$ C. det (A) = $a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{31}$. D. $\lvert AA'\rvert \neq \lvert A\rvert^2$ Choose the correct answer from the options given below:

The system of equation $x-y+z=4, x-2y-2z=9, 2x+y+\alpha z=1$ has unique solution then value(/s) of $\alpha$ is (/are) (where $\mathbb{R}$ is set of real numbers)

For matrix $A = \begin{bmatrix} 2 & -3 \\ 3 & 4 \end{bmatrix}$ the value of $A^2 - 6A$ is equal to:(where I is an identity matrix of order 2)

For a matrix $A = \begin{bmatrix} a & -1 & b \\ -2 & 1 & -1 \\ 1 & 0 & 2 \end{bmatrix}$, $C_{32} = -1$ and $C_{33} = 3$, where $C_{ij}$ is the cofactor of element $a_{ij}$ in $A$. Then

If $A = \begin{bmatrix} 1 & a \\ b & 2 \end{bmatrix}$ and $B = \begin{bmatrix} a & 1 \\ 2 & -3b \end{bmatrix}$ such that $\lvert AB\rvert = \lvert 2B\rvert$ and $ab \neq -\dfrac{2}{3}$ then: A. $\lvert A\rvert = 2$ B. $\lvert A\rvert = 4$ C. $\lvert B\rvert = 2$ D. $\lvert B\rvert = 4$ Choose the correct answer from the options given below:

The value of $\begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix}$ is equal to

If $f(x) = \begin{vmatrix} 0 & x+a & x+b \\ x-a & 0 & x+c \\ x-b & x-c & 0 \end{vmatrix}$ then the value of $f(0)$ is equal to:

The number of diagonal matrices of order 3 with elements either 1 or 2, is:

If $A^2 = 8A + kI, A = \begin{bmatrix} 1 & 0 \\ -1 & 7 \end{bmatrix}, I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ then value of $k$ is:

Match List-I with List-II | List-I | List-II | |---|---| | (A) If $A = [a_{ij}]_{n \times n}$, where $a_{ij} = \begin{cases}0, & i \ne j \\ k, & i=j, k\ne 0\end{cases}$ then A is ___ | (I) Symmetric matrix | | (B) If $A = [a_{ij}]_{2\times 2}$, where $a_{ij} = \begin{cases}1, & i \ne j \\ 0, & i=j\end{cases}$, then $A^2$ is ___ | (II) Skew-Symmetric matrix | | (C) If A and B are matrices of the same order, then $(AB^T - BA^T)$ is ___ | (III) Identity matrix | | (D) If A and B are symmetric matrices of the same order, then $(AB+BA)$ is ___ | (IV) Scalar matrix | Choose the correct answer from the options given below:

If $A = \begin{bmatrix}0 & a & -3 \\ 2 & 0 & -1 \\ b & 1 & 0\end{bmatrix}$ is a skew-symmetric matrix, then

If $\begin{bmatrix}3 & 1\\2 & 1\end{bmatrix}P\begin{bmatrix}2 & 1\\3 & 2\end{bmatrix}=\begin{bmatrix}1 & -1\\0 & 1\end{bmatrix}$ then matrix P is

If $A=\begin{bmatrix}0 & b-a & c-a\\a-b & 0 & c-b\\a-c & b-c & 0\end{bmatrix}$ then the value of $\lvert A \rvert$ is

Let A be the area of the triangle with vertices (3, 8), (5, -1) and $(k, 2)$, $k \in \mathbb{Z}$ (set of integers). If $2A = 75$, then k is equal to:

If $A=\begin{bmatrix}k & 2\\2 & k\end{bmatrix}$ such that $\lvert A^3 \rvert = 125$ then which of the following statements are TRUE? (A) $\lvert k.adj(3A) \rvert = 405$ (B) $\lvert k.adj(3A) \rvert = 2025$ (C) $\lvert adj(3A) \rvert = 95$ (D) $\lvert adj(3A) \rvert = 45$ Choose the correct answer from the options given below:

If $A=\begin{bmatrix}5 & 0 & 0\\0 & 5 & 0\\0 & 0 & 5\end{bmatrix}$ such that $A^5=\lambda^2 A$, then the value of $\lambda$ is

If $A=\begin{bmatrix}x+4 & 3\\2 & 1\end{bmatrix}, B=\begin{bmatrix}5 & 2x-1\\2 & x\end{bmatrix}$ and $\lvert adj(AB) \rvert=32$, then the value of x is:

Match List-I with List-II | List-I | List-II | |---|---| | Matrix A | $\left\vert \text{adj } A \right\vert^2$ | | (A) $\begin{bmatrix} 3 & 7 \\ 2 & 8 \end{bmatrix}$ | (I) $121$ | | (B) $\begin{bmatrix} 5 & -1 \\ 2 & 4 \end{bmatrix}$ | (II) $144$ | | (C) $\begin{bmatrix} 3 & 6 \\ 5 & 14 \end{bmatrix}$ | (III) $484$ | | (D) $\begin{bmatrix} 7 & 5 \\ 2 & 3 \end{bmatrix}$ | (IV) $100$ | Choose the correct answer from the options given below:

If $A=\begin{bmatrix}1 & 2 & 2\\2 & 1 & -2\\\alpha & 2 & \beta\end{bmatrix}$ and $AA^T=9I$, where I is the identity matrix of order 3, then $(\beta-\alpha)$ equals:

Which of the following statements are true? (A) A matrix which is both symmetric as well as skew-symmetric is a null matrix. (B) All positive integral powers of a symmetric matrix are symmetric. (C) If A is a square matrix, then $A+A^T$ is a symmetric matrix. (D) All positive even integral powers of a skew-symmetric matrix are skew-symmetric. Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) If $A=\begin{bmatrix}2 & -5\\3 & 4\end{bmatrix}$, then adj A = | (I) $\begin{bmatrix}4 & -5\\-2 & -3\end{bmatrix}$ | | (B) If $A=\begin{bmatrix}2 & 3\\5 & -2\end{bmatrix}$, then $A^{-1}=$ | (II) $\begin{bmatrix}\frac{1}{17} & \frac{-5}{17}\\\frac{3}{17} & \frac{2}{17}\end{bmatrix}$ | | (C) If $A=\begin{bmatrix}-3 & 5\\2 & 4\end{bmatrix}$, then adj A = | (III) $\begin{bmatrix}4 & 5\\-3 & 2\end{bmatrix}$ | | (D) If $A=\begin{bmatrix}2 & 5\\-3 & 1\end{bmatrix}$, then $A^{-1}=$ | (IV) $\begin{bmatrix}\frac{2}{19} & \frac{3}{19}\\\frac{5}{19} & \frac{-2}{19}\end{bmatrix}$ | Choose the correct answer from the options given below:

The value of $\begin{vmatrix} 0 & xy^2 & xz^2 \\ x^2y & 0 & yz^2 \\ x^2z & zy^2 & 0 \end{vmatrix}$ is:

If $A=\begin{bmatrix}4 & 1\\2 & 3\end{bmatrix}$ and $B=\begin{bmatrix}-2 & 3\\1 & 2\end{bmatrix}$ such that $2B-3A+X=0$ then $X$ is

Let $A$ be a matrix given by $A=[a_{ij}]_{3\times3}$, $a_{ij}=i+j$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) Number of the elements in $A$ | (I) 3 | | (B) Sum of the diagonal elements | (II) 9 | | (C) $a_{31}+a_{32}-a_{33}$ | (III) -1 | | (D) Cofactor of $a_{13}$ | (IV) 12 | Choose the correct answer from the options given below:

For every square matrix $A$ of order $n\times n$, which of the following statements are correct ? (A) $(adjA)A = A(adjA) = |A|I$, where $I$ is identity matrix of order n. (B) $|kA|=k^n|A|$, k is any scalar. (C) $|adjA|=|A|^n$ (D) $A^2=A$ Choose the correct answer from the options given below:

If $adjA=\begin{bmatrix}7 & -3 & 2\\3 & 0 & -3\\-1 & 3 & 1\end{bmatrix}$, then $A^{-1}$ is:

If $A=\begin{bmatrix}1 & 1 & 1\\1 & 0 & 3\\1 & -2 & 1\end{bmatrix}$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) $\lvert A\rvert$ | (I) 48 | | (B) $\lvert 2A\rvert$ | (II) 36 | | (C) $\left\lvert\operatorname{adj}(A)\right\rvert$ | (III) 12 | | (D) $2\lvert A\rvert$ | (IV) 6 | Choose the correct answer from the options given below:

If A is a square matrix such that $A^2=A$ and $I$ is identity matrix of same order, then $(2I+A)^3-19A$ is equal to:

If $A=\begin{bmatrix}3 & -2\\4 & -2\end{bmatrix}$ and $I=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$ such that $A^2-A+\lambda I=0$, then value of '$\lambda$' is:

If $A=[a_{ij}]_{3\times3}=\begin{bmatrix}2 & 3 & -1\\1 & 2 & 3\\0 & -1 & 1\end{bmatrix}$ and $B=[b_{ij}]_{3\times2}=\begin{bmatrix}3 & -2\\1 & 4\\1 & 2\end{bmatrix}$ then the value of $a_{12}b_{21}+a_{32}b_{31}$ is

If $XA=(I-A)^2$ where $X, A$ and $I$ are $2\times2$ matrices and $A=\begin{bmatrix}1 & 2\\1 & 4\end{bmatrix}$, then $|X|=$

If the matrix $A=\begin{bmatrix}0 & -1 & 3x\\1 & y & -5\\-6 & 5 & 0\end{bmatrix}$ is skew-symmetric, then the value of $(5x-y)$ is

If matrix $A=\begin{bmatrix}1 & 4 & 5\\3 & 2 & 6\\0 & 1 & 0\end{bmatrix}$, then Match List-I with List-II | List-I | List-II | |---|---| | (A) $\left\vert A \right\vert =$ | (I) $\dfrac{1}{9}$ | | (B) $\left\vert A^{-1} \right\vert =$ | (II) $81$ | | (C) $\left\vert \text{adj } A \right\vert =$ | (III) $9$ | | (D) $\left\vert \text{adj } (\text{adj } A) \right\vert =$ | (IV) $6561$ | Choose the correct answer from the options given below:

Which of the following are correct? (A) $A=\begin{bmatrix}-9 & 4 & -3\\-1 & 0 & 4\\2 & 2 & 0\end{bmatrix}$ ; $a_{22}+a_{32}=-9$ where $a_{ij}$ denotes the $(i,j)^{th}$ element of matrix A. (B) $A=\begin{bmatrix}2 & -2\\-2 & 2\end{bmatrix}$; $A^2=pA$, then $p=4$ (C) $\begin{bmatrix}8 & 6\\6 & 16\end{bmatrix}$ is a symmetric matrix (D) $\begin{bmatrix}0 & -5 & 3\\5 & 0 & -7\\-3 & 7 & 0\end{bmatrix}$ is a symmetric matrix Choose the correct answer from the options given below:

If $A=\begin{bmatrix}\lambda & 0 & 0\\0 & \lambda & 0\\0 & 0 & \lambda\end{bmatrix}$, $\lambda\ne0$, then $|adjA|$ is equal to

The matrix X for which $\begin{bmatrix}5 & 4\\1 & 1\end{bmatrix}X=\begin{bmatrix}1 & -2\\1 & 3\end{bmatrix}$ is

The number of all possible matrices of order $3 \times 3$ with each entry $-1, 0$ or $1$ is

Let A be a square matrix of order 3 and $|A| = -3$, then $|adj(3A)|$ is equal to:

Let A, B, C be square matrices of equal orders and I be the identity matrix of same order, Then which of the following are TRUE? A. $(AB)C = A(BC)$ B. $AB = BA$ C. $A(B+C) = AB + AC$ D. $AI = IA = A$ Choose the correct answer from the options given below:

Let $I = \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$ and $A = \begin{bmatrix}1 & -2\\2 & 4\end{bmatrix}$ be two matrices such that $8A^{-1} = kI - A, k \in \mathbb{R}$, then the value of $k$ is:

If the system of equation $\lambda x + 3y - z = 4$ $x + 2y + z = 20$ $-\lambda x + y + 2z = -5$ does not possess a unique solution, then

If A is a matrix of order $3 \times 3$ such that $|A| = 4$ and $|k \cdot adj\, A| = 32|A|, k \in \mathbb{R}$; then the value of k is:

The matrix $A = \begin{bmatrix}-3 & 0 & 0\\0 & -3 & 0\\0 & 0 & -3\end{bmatrix}$ is a A. Symmetric matrix B. Scalar matrix C. Diagonal matrix D. Skew-symmetric matrix Choose the correct answer from the options given below:

The value of $\begin{vmatrix}2 & 3 & 4\\5 & 6 & 8\\6x & 9x & 12x\end{vmatrix}$ is

If $A = \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ such that $A^2 + \beta A + \gamma I = 0$ then $\beta + \gamma$ is equal to: (where $I$ is an identity matrix of order 2)

If $A = \begin{bmatrix}-2 & 2 & 0\\3 & 1 & 4\end{bmatrix}$ and $B = \begin{bmatrix}2 & 0 & -2\\7 & 1 & 6\end{bmatrix}$ are two matrices such that $-3A + 5B + 2C = 0$ then matrix C is equal to:

If $\begin{vmatrix}x-2 & 2x-3 & 3x-4\\x-4 & 2x-9 & 3x-16\\x-8 & 2x-27 & 3x-64\end{vmatrix} = 0$ then the value of x is:

If A is a square matrix of order 3 with $|A| = 4$, then the value of $|2\,adj\,A|$ is:

If $A = \begin{bmatrix}1 & -1\\2 & -1\end{bmatrix}, B = \begin{bmatrix}x & 1\\y & -1\end{bmatrix}$ and $(A+B)^2 = A^2 + B^2$, then which of the following are correct? A. $x=1$ B. $x=4$ C. $y=1$ D. $y=4$ Choose the correct answer from the options given below:

If matrix $A = \begin{bmatrix}12 & 7 & 9\\6 & 3 & 2\\5 & 1 & 8\end{bmatrix}$, then Match the LIST-I with LIST-II | | LIST-I | | LIST-II | |---|---|---|---| | A. | $\vert A \vert$ | I. | $38$ | | B. | Cofactor of $a_{31}$ | II. | $-23$ | | C. | Minor of $a_{23}$ | III. | $-83$ | | D. | Cofactor of $a_{12}$ | IV. | $-13$ | Choose the correct answer from the options given below:

If $A = \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\end{bmatrix}$ and $B = \begin{bmatrix}1 & 2\\3 & 4\\5 & 6\end{bmatrix}$, then $A + B$ is equal to

For a matrix $A = \begin{bmatrix} 1 & 4 & 5 \\ 3 & 2 & 6 \\ 0 & 1 & 0 \end{bmatrix}$, where $I$ is identity matrix of order 3 Match List-I with List-II | List-I | List-II | |---|---| | (A) $\left\vert A \right\vert$ | (I) $72$ | | (B) $(\text{adj}A)A$ | (II) $18$ | | (C) $\left\vert 2A \right\vert$ | (III) $9$ | | (D) $2\left\vert A \right\vert$ | (IV) $9I$ | Choose the correct answer from the options given below:

For any 3 ×3 matrix $A$, if $A(adj\,A) = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}$, then $2|A|$ is equal to

If $A = \begin{bmatrix} 4 & -1 & \lambda \\ 2 & 0 & 6 \\ 1 & -2 & 7 \end{bmatrix}$, then $\frac{1}{2}A^{-1}$ exist if

If $[1 \quad 2 \quad 1] \begin{bmatrix} -1 & 0 & 1 \\ 1 & 2 & -1 \\ 0 & 1 & 3 \end{bmatrix} \begin{bmatrix} 2x \\ x \\ -7 \end{bmatrix} = [0]$, then the value of $x$ is

If $A = [a_{ij}]_{3 \times 3}$ and $a_{ij} = i - j$, then which of the following statements are TRUE? (A). $A$ is scalar matrix (B). $A$ is skew symmetric matrix (C). $|A| = 0$ (D). $a_{21} + a_{12} = 0$ Choose the correct answer from the options given below:

If $\begin{vmatrix} 2+x & y & z \\ x & 2+y & z \\ x & y & 2+z \end{vmatrix} = 0$, then the value of $x+y+z$ is

If $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 3 & -1 \end{bmatrix}$, then $A^2$ is equal to

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ m & 2 & n \end{bmatrix}$ is a matrix, such that $AA^T = 9I_3$, then the values of $m$ and $n$ are (where $I_3$ is an identity matrix of order 3)

For any invertible matrix $A$, which of the following is NOT true.

If $\Delta = \begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix}$, Then Match List-I with List-II | List-I | List-II | |---|---| | Values of $a, b$ & $c$ | value of $\Delta$ | | (A). $a=b=c=1$ | (I). 2 | | (B). $a=-1,b=1,c=1$ | (II). 0 | | (C). $a=-1,b=-1,c=1$ | (III). -2 | | (D). $a=-1,b=-1,c=-1$ | (IV). 4 | Choose the correct answer from the options given below:

If $A$ and $B$ are matrices of the same order, then $AB^T - BA^T$ is a

Which of the following systems has only a trivial solution?

If $\begin{vmatrix} 2 & 3 & 2 \\ x & x & x \\ 4 & 9 & 1 \end{vmatrix} + \begin{vmatrix} 3 & 2 \\ 4 & 5 \end{vmatrix} = 0$, then the value of $x$ is

If the matrix $\begin{bmatrix} z & x-y+2 & x+4y-3 \\ x+y & 0 & a-3 \\ 2x-y+3 & a-1 & 0 \end{bmatrix}$ is skew-symmetric, then the value of $(a+x+y+z)$ is:

Which of the following matrices are invertible? (A) $\begin{bmatrix} 2 & 2 \\ 3 & 3 \end{bmatrix}$ (B) $\begin{bmatrix} 2 & 3 \\ 3 & 2 \end{bmatrix}$ (C) $\begin{bmatrix} 3 & 2 \\ 2 & 3 \end{bmatrix}$ (D) $\begin{bmatrix} 3 & 3 \\ 3 & 3 \end{bmatrix}$ Choose the correct answer from the options given below:

If A is a square matrix $\begin{bmatrix} 5 & -5 \\ -5 & 5 \end{bmatrix}$ such that $A^2 = p\,A$, then the value of p is,

If A is a square matrix of order $n$, then Match the LIST-I with LIST-II | | LIST-I | | LIST-II | |---|---|---|---| | A. | $\vert A(\text{adj }A) \vert$ | I. | $\vert A \vert^{n-1}$ | | B. | $\text{adj}(\text{adj }A)$ | II. | $\vert A \vert I$, where $I$ is the identity matrix of order $n$ | | C. | $\vert \text{adj }A \vert$ | III. | $\vert A \vert^{n-2} A$ | | D. | $A(\text{adj }A)$ | IV. | $\vert A \vert^n$ | Choose the correct answer from the options given below:

If A and B are two square matrices of order 'n', where $AB \neq BA$, then

A matrix which is both symmetric and skew symmetric is a

Let A be a square matrix of order 3 and I is an identity matrix of order 3. If $|A| = 3$, the value of $|A(4I)|$ is

The value of $\begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix}$ is:

If $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$, then $A^{26}$ is equal to

Match the LIST-I with LIST-II | | LIST-I<br>Characteristic of square matrix | | LIST-II<br>Type of matrix | |---|---|---|---| | A. | A is a square matrix such that $A + A^T = 0$. | I. | Identity matrix or scalar matrix | | B. | B is a square matrix such that $B^2 = B$ | II. | Diagonal matrix | | C. | C is a square matrix with all equal diagonal element and all the non-diagonal elements are zero. | III. | Skew symmetric matrix | | D. | D is a matrix such that all the non-diagonal elements are zero. | IV. | Idempotent matrix | Choose the correct answer from the options given below:

The value of $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$ is negative, if

If A and B are square matrices of same order such that, $AB=BA$, $A^{-1} = A^2$ and $B^{-1} = B^3$, then the value of $(AB)^{2040}$, is

If $\begin{vmatrix} 1+x^2 & 1 & 1 \\ 1 & 1+y^2 & 1 \\ 1 & 1 & 1+z^2 \end{vmatrix} = k\left(1 + \frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}\right)$ then $k$ is equal to

The value of $\begin{vmatrix} \log_3 512 & \log_4 3 \\ \log_3 8 & \log_4 9 \end{vmatrix}$ is:

If $x \in N$(Set of Natural Numbers) and $\begin{vmatrix} x+3 & -2 \\ -3x & 2x \end{vmatrix} = 8$, then the value of $x$ is

If $f(x) = \begin{vmatrix} 1 & x & x^2 \\ x & x^2 & 1 \\ x^2 & 1 & x \end{vmatrix}$, then the value of $f(\sqrt[3]{3})$ is

Which of the following matrices are diagonal matrices? A. $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$ B. $\begin{bmatrix} 1 & 2 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$ C. $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ D. $\begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}$ Choose the correct answer from the options given below:

If A and B are invertible matrices of the same order, then, Match List-I with List-II | LIST-I | LIST-II | |---|---| | A. $\left\vert A^T \right\vert$ | I. $B^{-1}A^{-1}$ | | B. $A(\text{adj }A)$ | II. $\text{adj }A$ | | C. $A^{-1}\left\vert A \right\vert$ | III. $(\text{adj }A)A$ | | D. $(AB)^{-1}$ | IV. $\left\vert A \right\vert$ | Choose the correct answer from the options given below:

If A is a square matrix such that $A^2 = I$, then $(A+I)^3 + (A-I)^3 - 7A$ is equal to