The value of $\lambda$ for which the following system of equations has no solution is

$x+y+2z=4$

$2x+y+3z=5$

$x+3y+\lambda z=7$

If $A = \begin{bmatrix} 7 & 3 \\ 5 & -7 \end{bmatrix}$ be such that $A^{-1} = kA$, then $k$ equals

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 :

Let P and Q be any two invertible matrices of the same order. Then

Match List-I with List-II

Choose the correct answer from the options given below:

Which of the following statements are true?

(A) A matrix which is both symmetric as well as skew-symmetric is a null matrix.

(B) All positive integral powers of a symmetric matrix are symmetric.

(C) If A is a square matrix, then $A+A^T$ is a symmetric matrix.

(D) All positive even integral powers of a skew-symmetric matrix are skew-symmetric.

Choose the correct answer from the options given below:

Let $A=\begin{bmatrix}a & 0 \\ b & 0\end{bmatrix}$ and $A^2=\begin{bmatrix}4 & c \\ 2 & 0\end{bmatrix}$, $a,b,c \in \mathbb{Z}$. Then the minimum value of $a+b+c$ is:

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