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.	$\vert A \vert^n$

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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 values of $x, y, z$ and $w$ such that $\begin{bmatrix} x-y & 2z+w \\ 2x-y & 2x+w \end{bmatrix} = \begin{bmatrix} 5 & 3 \\ 12 & 15 \end{bmatrix}$ are given by

If $A = \begin{bmatrix} 5 & 2 \\ 4 & 3 \end{bmatrix}$ is a given matrix, then which of the following statements are correct?

(A) $|A| = 7$

(B) minor of $3 = -5$

(C) co-factor of $2 = -4$

(D) $adj(A) = \begin{bmatrix} 3 & -2 \\ -4 & 5 \end{bmatrix}$

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If $A^T = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and $A^{-1} = \begin{bmatrix} p & q \\ r & s \end{bmatrix}$, then

Let $A = \begin{bmatrix} 2 & -3 & 4 \\ 0 & 1 & 5 \\ -4 & 2 & 3 \end{bmatrix}$ and $a_{ij}$ be any element of matrix A, i, j ∈ {1,2,3}, then which of the following are TRUE?

(A) Minor of $a_{23} = 16$

(B) Minor of $a_{23} = -8$

(C) Cofactor of $a_{23} = -16$

(D) Cofactor of $a_{23} = 8$

(E) Cofactor of $a_{13} = 4$

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The system of equations in 3 variables $x, y$ and $z$ as $x+y=5$, $y+az=3$, $x+by=8$ has a unique solution if the value of $a$ and $b$ are

(A) $a=1, b=1$

(B) $a=0, b=1$

(C) $a \in (-\infty,0)$ and $b\neq1$

(D) $a\neq0$ and $b\in(3,4)$

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