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:

Let A be a non singular square matrix of $2 \times 2$. Then, $|adj\ A|$ is equal to

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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Which of the following statements are correct?

(A) If A is a square matrix, then $|A^2| = |A|^2$.

(B) If A and B are square matrices of the same order, then det (AB) = det (A) + det (B).

(C) If A is a square matrix of order 3 and $|A|=2$, then the value of $|-3A|$ is 54.

(D) If the matrix $\begin{bmatrix} 5 -x & x -1 \\ 3 &5 \end{bmatrix}$ is singular, then the value of x is 7/2.

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

(A) $AB^T+B^TA$ is a skew-symmetric matrix.

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

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

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

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If $y = -4$ is a root of $\begin{vmatrix} y & 2 & 3 \\ 1 & y & 1 \\ 3 & 2 & y \end{vmatrix} = 0$, then the product of the other two roots is

The inverse of the matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ is: