If A is a $2 \times 3$ matrix and B is a matrix such that $A^{T}B$ and $BA^{T}$ are both defined. Then order of B is

If $\begin{vmatrix} (a-x)^2 & (a-y)^2 & (a-z)^2 \\ (b-x)^2 & (b-y)^2 & (b-z)^2 \\ (c-x)^2 & (c-y)^2 & (c-z)^2 \end{vmatrix} = \lambda(a-b)(b-c)(c-a) \cdot (x-y)(y-z)(z-x)$

then the value of $\lambda$ is:

For the system of equations AX = B, which of the following is correct?

The value of the determinant $\Delta = \begin{vmatrix} 1! & 2! & 3! \\ 2! & 3! & 4! \\ 3! & 4! & 5! \end{vmatrix}$ is :

The system of equation $x-y+z=4, x-2y-2z=9, 2x+y+\alpha z=1$ has unique solution then value(/s) of $\alpha$ is (/are) (where $\mathbb{R}$ is set of real numbers)

If $A = [a_{ij}]$ is a square matrix of order 2 such that $a_{ij} = \begin{cases} 2, & \text{when } i \neq j \\ 0, & \text{when } i = j \end{cases}$, then det $(A^2)$ is:

Let the matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$. Then which of the following are true?

(A) adj $A = \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}$

(B) det $(A) = 5$

(C) det (adjA) = 25

(D) If $A^3 = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $a + b = c + d$

Choose the correct answer from the options given below: