Which of the following statements is (are) true? (A) $B^T AB$ is a skew-symmetric matrix if A is a symmetric matrix (B) $B^T AB$ is a symmetric matrix if A is a symmetric matrix (C) $B^T AB$ is a symmetric matrix if A is a skew-symmetric matrix (D) $B^T AB$ is a skew-symmetric matrix if B is a skew-symmetric matrix (E) $B^T AB$ is a symmetric matrix if B is a symmetric matrix Choose the correct answer from the options given below:

If $A = \begin{bmatrix} 5 & 6 \\ 3 & 2 \end{bmatrix}$ then which of the following is correct?

(A) $|A|$ is positive

(B) $|adj\ A| = -8$

(C) Cofactor of 3 is 6

(D) $|2A| = -32$

Choose the correct answer from the options given below:

If $a,b,c$ are the roots of $x^3+px+q=0$, then the value of $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$ is equal to

$\begin{vmatrix} a & b-c & c+b \\ a+c & b & c-a \\ a-b & a+b & c \end{vmatrix}$ is equal to:

If A and B are invertible matrices of order 3, $|A| = 2$ and $|(AB)^{-1}| = -\frac{1}{6}$, then the value of $|B|$ is :

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

$x + y + z = 1$

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

$x - y + kz = 5$

Which of the following statements are correct?

(A) Inverse of a matrix, if it exists, is unique

(B) $(kA)' = -kA'$ (where k is any real number)

(C) For an invertible matrix $A$, $(A^{-1})^{-1} = A$

(D) For an invertible matrix $A$, $(A')^{-1} = (A^{-1})'$

Choose the correct answer from the options given below: