If $\begin{vmatrix} 1+x^2 & 1 & 1 \\ 1 & 1+y^2 & 1 \\ 1 & 1 & 1+z^2 \end{vmatrix} = k\left(1 + \frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}\right)$ then $k$ is equal to

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 $A = [a_{ij}]_{2×2}$ where $a_{ij} = \begin{cases} 1, & i \neq j \\ 0, & i = j \end{cases}$ and $I$ is the identity matrix of order 2, then $(A^2 - 3A + 4I)$ is

(A) Symmetric Matrix

(B) Skew-symmetric Matrix

(C) Non-singular Matrix

(D) Square Matrix

Choose the correct answer from the options given below:

If A and B are two non-singular matrices of order n, then which of the following statement/statements is/are not correct?

(A) AB is non-singular.

(B) AB is singular.

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

(D) $(AB)^{-1}$ does not exist.

Choose the correct answer from the options given below:

If A is a skew-symmetric matrix then, the matrix $B^TAB$ is:-

If $\left[\begin{array}{cc}5 x+8 & 7 \\ y+3 & 10 x+12\end{array}\right]=\left[\begin{array}{cc}2 & 3 y+1 \\ 5 & 0\end{array}\right]$, then the value of $5 x+3 y$ is equal to :

If $A=\begin{bmatrix} 2 & 1 \\ 7 & 5 \end{bmatrix}$, then the value of $|A(adj A)|$ is