Let $A = \begin{bmatrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ \lambda & 2 & -3 \end{bmatrix}$. If $A^{-1}$ does not exist, then $\lambda =$

If $C_{ij}$ represents the cofactor of element $a_{ij}$ of the matrix $A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & 0 \\ 4 & 1 & 5 \end{bmatrix}$ then the value of $C_{23} + C_{31} - C_{22}$ is

If $A = \begin{bmatrix} a & 4 & -5 \\ d & b & -6 \\ 5 & e & c \end{bmatrix}$ is a skew symmetric matrix, then value of $a + b + c + d + e$ is equal to

If $A^T = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}$, then the matrix $(A + 2B)^T$ is

If $A = \begin{bmatrix} 0 & 1 & -3 \\ -1 & 0 & 5 \\ 3 & -5 & 0 \end{bmatrix}$, then the value of $|A^{2025}|$ is

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:

Let A = [aᵢⱼ]ₙₓₙ and B = [bᵢⱼ]ₙₓₙ. Then which of the following is/are true?

(A) AB = BA

(B) (AB)⁻¹ = B⁻¹ A⁻¹

(C) $(AB)^T = B^T A^T$

(D) AB = 0 ⇒ A = 0 or B = 0

Choose the correct answer from the options given below: