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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

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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) $A^2=A$

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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

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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

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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

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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. $AI = IA = A$

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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

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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. $y=4$

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If matrix $A = \begin{bmatrix}12 & 7 & 9\\6 & 3 & 2\\5 & 1 & 8\end{bmatrix}$, then

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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