The sum of the products of elements of any row with the cofactors of corresponding elements is equal to :

Let A and B be square matrices of order 3 and k is a constant. If $|A| \neq 0$ and $|B| \neq 0$, where $|A|$ represents the determinant of A, then which of the following statements are true ? (where A' denotes the transpose of matrix A)

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

(B) $(A+B)' = A' \times B'$

(C) $(AB)' = A'B'$

(D) $|kA| = k^3 |A|$

(E) $|A'| = |A|$

Choose the correct answer from the options given below :

If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then the value of $A^2 - 5A + 6I$ is

If $A = \begin{bmatrix} 5 & 3 \\ 2 & 4 \end{bmatrix}$, then the matrix $A^2 - 6A + 14$ I is (where I is an identity matrix of order 2)

Let A be a square matrix of order n, then which of the following are TRUE?

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

(B) $|A. adj A| = |A|^n$

(C) $A. (adj A) = |A|$

(D) $|KA| = K|A|$

(E) $|A^{-1}| = \frac{1}{|A|}, |A| \neq 0$

Choose the correct answer from the options given below:

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 $\begin{bmatrix}-2 & 0\\ x & 4\end{bmatrix}+3\begin{bmatrix}y & 0\\ 2 & -1\end{bmatrix}=I$, where $I$ is $2\times2$ unit matrix, then the value of $x$ and $y$ are: