If $x, y$ & $z$ are non zero real numbers, the inverse of matrix $A = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ is :

If $A$ is a square matrix of order $n \times n$, then which of the following are correct?

(A) $|kA| = k^n|A|$, $k$ is a constant

(B) $|k\ adjA| = k^n|adjA|^{n-1}$, $k$ is a constant

(C) $|adj\ adjA| = |A|^{(n-1)^2}$

(D) $A(adjA) = |A|I$, $I$ is an identity matrix of order $n$

Choose the correct answer from the options given below:

For $I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, if $X$ and $Y$ are square matrices of order $2$ such that $X Y=X$ and $Y X=Y$, then $\left(\mathrm{Y}^{2}+2 \mathrm{Y}\right)$ equals to :

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 :

Let A be the area of the triangle with vertices (3, 8), (5, -1) and $(k, 2)$, $k \in \mathbb{Z}$ (set of integers). If $2A = 75$, then k is equal to:

Let $A=[a_{ij}]$ be a matrix of order $3 \times 4$ such that $a_{ij}=\begin{cases}2i-3j, & i \ge j \\ 3i+j, & i<j\end{cases}$ then the sum of all the elements of the first two rows of this matrix is:

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