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. $\left\vert A \right\vert$

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Let A = PQ. The elementary operation on A, that produces the same effect as it does on applying on P and keeping Q unchanged is :

(A) $R_i \leftrightarrow R_j$

(B) $R_i \to R_i + KR_j$

(C) $C_i \to KC_i$

(D) $C_i \to C_i + KC_j$

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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:

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. $\lvert AA'\rvert \neq \lvert A\rvert^2$

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The set of values of K for which the system of equations $\begin{bmatrix} 2 & 3 & 1 \\ 4 & 5 & 0 \\ 1 & K & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \\ 7 \end{bmatrix}$ gives a unique solution is :

If a matrix $A=\begin{bmatrix} 3 & -5 & 7 \\ 5 & 2 & 4 \\ -7 & -4 & 9 \end{bmatrix}$ is expressed as the sum of a symmetric matrix P and a skew symmetric matrix Q, then symmetric matrix P will be

Consider the system of equations $x+y+z=2, 2x+3y+2z=5, x+2y+\lambda z=\mu$ then which of the following statements are TRUE?

A. The system of equations has unique solution if $\lambda=2$ and $\mu\in\mathbb R$ (set of real number)

B. The system of equations is inconsistent if $\lambda=1$ and $\mu=1$

C. The system of equations is consistent if $\lambda=1$ and $\mu=1$

D. The system of equations is consistent if $\lambda=1$ and $\mu=3$

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