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

The matrix $X$ in the equation $AX = B$, such that $A = \begin{bmatrix} 1 & 3 \\ 0 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix}$ is given by

Let A be any skew- symmetric matrix (where $A^T$ is Transpose of matrix A). Then which of the following statements are correct?

(A) $A^2$ is a symmetric matrix

(B) $A^2$ is a skew- symmetric matrix

(C) $A^T A = -A^2$

(D) $A^T A - AA^T = O$

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If $\begin{bmatrix} x-y & 0 \\ x+y & 1 \end{bmatrix}$ is an identity matrix and $\begin{bmatrix} x & y \\ z & x \end{bmatrix}$ is a singular matrix then:

If the area of a triangle with vertices $(-3,0)$, $(3, 0)$ and $(0, k)$ is 9 sq. units, then k equals

If $x, y$ and $z$ are non-zero distinct numbers, then $\begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix}$ is equal to

The system of linear equations $kx + 5y = 5$, $2x + 3y = 5$ will be consistent if

If $\begin{vmatrix} 1 & \cos \theta & 0 \\ \sin \theta & 1 & \cos \theta \\ |\cos \theta & 1 & -\sin \theta| \end{vmatrix} = A\sin \theta + B\cos \theta + C\sin \theta\cos \theta$ then:

If $A = \begin{bmatrix} 4 & 3 \\ 2 & -1 \\ 1 & 0 \end{bmatrix}$ and $B^T = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & -1 \end{bmatrix}$, then $A - B$ is equal to

The number of all possible matrices of order 3 with each entry either 0 or 1 is:

Match List-I with List-II

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If $A$ and $B$ are square matrices of the same order, then which of the following statements are correct?

(A) $|A^{-1}| = |A|^{-1}$

(B) $adj(A) = |A|A^{-1}$

(C) $(A + B)^{-1} = B^{-1} + A^{-1}$

(D) $(AB)^{-1} = B^{-1}A^{-1}$

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If A and B are two non-singular matrices of order n, then which of the following statement/statements is/are not correct?

(A) AB is non-singular.

(B) AB is singular.

(C) $(AB)^{-1} = A^{-1}B^{-1}$

(D) $(AB)^{-1}$ does not exist.

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If the matrix $A = \begin{bmatrix} \alpha & \beta & \gamma \\ 0 & 0 & 2 \\ 3 & -2 & 0 \end{bmatrix}$ is a skew symmetric matrix, then the value of $(\alpha + \beta + \gamma)^2$ is:

If $A = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix}$, $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ and $A² = KA - 2I$, then the value of K is

Let $A = [a_{ij}]_{3×2}$ and $B = [b_{ij}]_{3×4}$ be two matrices. Then the order of the matrix $(A^T . B)^T$ is:

Let $M_{ij}$ and $A_{ij}$ denote respectively minors and co-factors of the element in the $i^{th}$ row and $j^{th}$ column of the matrix $A = \begin{bmatrix} 1 & 2 & -1 \\ 3 & 2 & 3 \\ 4 & -1 & 0 \end{bmatrix}$. Then:

(A) $M_{32} = 6$

(B) $M_{23} = 9$

(C) $A_{32} = -6$

(D) $A_{23} = -9$

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The inverse of the matrix $A = \begin{bmatrix} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{8} \end{bmatrix}$ is

If $\begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix}$, where a, b and c are non zero constants, then the value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ is

If the matrix $A = \begin{bmatrix} 1 & 3 \\ 2 & 1 \end{bmatrix}$, then the value of det $(A^2 - 2A)$ is equal to

Let A be any square matrix of order n, then which of the following are true?

(A) $| \text{adj } A| = |A|^{n-1}$

(B) $|A^{-1}| = \frac{1}{|A|}$

(C) $| \text{adj } A| = |A|^n$

(D) $(A^T)^{-1} = (A^{-1})^T$

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Let the matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$. Then which of the following are true?

(A) adj $A = \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}$

(B) det $(A) = 5$

(C) det (adjA) = 25

(D) If $A^3 = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $a + b = c + d$

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For some constant 'k', if the system of linear equations

$2x - y + 3z = 1$

$x - 2y + z = 3$

$kx + y - z = 0$

has a unique solution, then

If $\begin{bmatrix} x+y & 4 \\ 1+z & y \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix}$, then

The system of equations

$x + ky = 0$

$3x + 5y = 0$

has infinitely many solutions, then k is equal to

If A is a square matrix of order 3 such that $A( {adj } A) = \begin{bmatrix} -2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -2 \end{bmatrix}$, then $|A|$ is equal to

If $\begin{bmatrix} 1 & 3 & 9 \\ 1 & x & x^2 \\ 4 & 6 & 9 \end{bmatrix}$ is singular matrix, where $x \in \mathbb{N}$ (where N set of natural number), then x is equal to

If $\begin{bmatrix} x-y & t \\ 2x-y & w \end{bmatrix} = \begin{bmatrix} -1 & 2 \\ 0 & 1 \end{bmatrix}$, then $2x + y + 3t + w$ is equal to

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

(A) AB-BA is a skew-symmetric matrix.

(B) AB+BA is a skew-symmetric matrix.

(C) $AB^T-BA^T$ is a skew-symmetric matrix.

(D) AB+BA is a symmetric matrix.

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If $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$, then the value of $A^{20}$ is:

If $A = \begin{bmatrix} 2 & 1 & 3 \\ 4 & -3 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} -2 & 3 \\ 4 & -5 \\ 1 & 2 \end{bmatrix}$, then which of the following statements are TRUE?

(A) AB is defined

(B) AB and BA both are defined and AB = I, where I is an identity matrix of order 2

(C) BA is defined

(D) AB and BA both are defined and AB = BA

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