Let $A$ and $B$ be square matrices of order 3, then det $[(A - A^T) + (B - B^T)]$ is equal to

The value of k for which the system of equations

$x + y + z = 1$

$x - ky + z = 1$

$x - y + z = 1$

has more than one solutions is

If $A = \begin{bmatrix} 2 & 0 & 3 \\ -1 & 1 & 3 \\ 0 & -4 & 0 \end{bmatrix}$, then the value of det (2A) is

If A is an invertible matrix, then which of the following statement(s) is/are TRUE?

(A) $|A^{-1}| = |A|$

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

(C) $A^{-1} = \frac{adj A}{|A|}$

(D) $(A^T)^{-1} = (A^{-1})^T$

Choose the correct answer from the options given below:

For what value of $k$, the following system have a unique solution? (where $\mathbb{R}$ is set of real numbers)

$x + y + z = 1$

$2x + 3y + 4z = 3$

$x - y + kz = 5$

Let $A = \begin{bmatrix} 152 & 105 & 3 \\ 149 & 25 & 35 \\ 2 & 1 & 0 \end{bmatrix}$. If $A_{ij}$ denotes the co-factor of an element $a_{ij}$ of the matrix A, then the value of $a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23}$ is equal to

If three points $A(a_1, b_1), B(a_2, b_2)$ and $C(a_3, b_3)$ are collinear and $D = \begin{vmatrix} a_1 & b_1 & 1 \\ a_2 & b_2 & 1 \\ a_3 & b_3 & 1 \end{vmatrix}$, then: