CUET Mathematics: AlgebraMatrices & Determinants. Free, no login required.

Let $A = [a_{ij}]_{n \times n}$ be a matrix, then match List-I with List-II

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If $A = \begin{bmatrix} 0 & 0 & \sqrt{7} \\ 0 & \sqrt{7} & 0 \\ \sqrt{7} & 0 & 0 \end{bmatrix}$, then $|\text{adj } A|$ is equal to

The system of equations $x + y - z = 1, 3x + y - 2z = 3, x - y + \lambda z = 1$ has infinite number of solutions if $\lambda$ is equal to

Let $A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$, then $(A^{-1})^T$ equals

Let $A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & -1 \\ 4 & 1 & 2 \end{bmatrix}$. $M_{ij}$ and $A_{ij}$ respectively denote the minor and cofactor of an element $a_{ij}$ of matrix $A = [a_{ij}]$

(A) $M_{23} = 6$

(B) $A_{22} = -8$

(C) $A_{13} = 7$

(D) $M_{32} = -5$

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If $A$ and $B$ are invertible matrices of order $3$ then match List-I with List-II

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If A and B are skew-symmetric matrices, then which of the following is not true?

Let $A = [a_{ij}]_{2 \times 4}$ and $B = [b_{ij}]_{4 \times 2}$, then $|3AB|$ is equal to

If $A = \begin{bmatrix} a & 1 & -1 \\ 0 & b & 4 \\ 4 & 4 & c \end{bmatrix}$ and $abc = 12$, $b = 4a$, then the value of $|A(adjA)|$ is:

If $A = \begin{bmatrix} 0 & 1 & 3 \\ 1 & 2 & x \\ 2 & 3 & 1 \end{bmatrix}$ and $A^{-1} = \begin{bmatrix} \frac{1}{2} & -4 & \frac{5}{2} \\ -\frac{1}{2} & 3 & -\frac{3}{2} \\ \frac{1}{2} & y & \frac{1}{2} \end{bmatrix}$, then the value of $8x + 5y$ is:

Let $A$ be a non singular matrix of order $n \times n$, Then $|\text{adj }(3A)|$ is equal to:

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

Assume $P$, $Q$, $R$ and $W$ are matrices of order $3 \times 3$, $a \times 4$, $b \times c$ and $d \times a$ respectively. If $PQ + WR$ is well defined, then the value of $ab + cd$ is:

If $A = [a_{ij}]$ is a square matrix of order 2 such that $a_{ij} = \begin{cases} 2, & \text{when } i \neq j \\ 0, & \text{when } i = j \end{cases}$, then det $(A^2)$ is:

Suppose that A, B and C are matrices of order $m \times n$, $n \times 5$ and $5 \times q$ respectively. The restriction on $m$, $n$ and $q$ so that AB-BC is defined are

If $A = \begin{bmatrix} -1 & 2 & 3x \\ 2y & 4 & -1 \\ 6 & -1 & 0 \end{bmatrix}$ is a symmetric matrix, then the value of $2x - y$ is:

The system of equation $2x + \lambda y = 8$, $\lambda x + 8y = 3$

has a unique solution if the value of $\lambda$ is (are):

If the area of a triangle whose vertices are (-1, 3), (1, -5) and (k, 2) where $k > 0$ is 30 sq. units, then the value of k is

If the system of equation

$x - y + z = 4$

$x - 2y - 2z = 9$

$2x + y + \lambda z = 1$

has a unique solution, then

If $A = \begin{bmatrix} 1 & 2 \\ 4 & -3 \end{bmatrix}$ and $f(x) = 2x^2 - 4x + 5$, the $f(A)$ is equal to

If $x \neq y \neq z$ then $\begin{vmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \end{vmatrix}$ is equal to

If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$ then $A^2 - 5A$ is equal to (where I is identity matrix of order 2)

Which of the following statement are correct?

(A) $A = [a_{ij}]_{n \times n}$ is a diagonal matrix if $a_{ij} = 0$ when $i = j$

(B) A square matrix $A = [a_{ij}]$ is called a symmetric matrix if $a_{ij} = a_{ji}$ for all $i, j$

(C) A square matrix $A = [a_{ij}]$ is called a skew-symmetric matrix if $a_{ij} = -a_{ji}$ for all $i, j$

(D) For every square matrix $A$, there exist an identity matrix of the same order such that $IA = AI = I$

Choose the correct answer from the options given below:

If A is a square matrix such that $A^2 = A$ then which of the following statements are TRUE ? (Where I is an identity matrix of same order as A)

(A) $(I+A)^4 = I + 15A$

(B) $(I+A)^2 = I + 3A$

(C) $(I+A)^6 = I + 30A$

(D) $(I+A)^3 = I + 7A$

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$A = \begin{bmatrix} 1/3 & 2 \\ 0 & 2x - 3 \end{bmatrix}$ & $B = \begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix}$ If $AB = I$ (Where I is an identity matrix of order 2) , then value of x is

The inverse of the matrix $\begin{bmatrix} 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6 \end{bmatrix}$ is

If matrix $A_p = \begin{bmatrix} p & p+1 \\ p-1 & p \end{bmatrix}_{p \in N}$ (where N is the set of natural numbers), then the value of $|A_1| + |A_2| + |A_3| + \dots + |A_{2025}|$ is:

The system of equations

$x - 3y - 8z = -10$

$2x + 5y + \lambda z = 13$

$3x + y - 4z = 0$

has infinite number of solutions if the value of $\lambda$ is equal to:

If A and B are symmetric matrices of the same order, then which of the following are true?

(A) AB - BA is a skew symmetric matrix

(B) AB is a symmetric matrix

(C) AB is a scalar matrix

(D) AB + BA is a symmetric matrix

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Let $A = [a_{ij}]$ is given by $A = \begin{bmatrix} 1 & -1 & 2 \\ 3 & 4 & -5 \\ 2 & -1 & 3 \end{bmatrix}$. Then the matrix $B = [b_{ij}]$, where $b_{ij}$ = Minor of $a_{ij}$ is: