$\begin{vmatrix} a+b & 1 & 0 \\ a^2-b^2 & a-b & 1 \\ a^3+b^3 & a^2+b^2+ab & a^2-b^2 \end{vmatrix} =$

If $A = \begin{bmatrix} 1 & -3 \\ 2 & 0 \end{bmatrix}$, then which of the following statement(s) is/are TRUE?

(A) Matrix A is non-singular

(B) $|3A| = 54$

(C) $|adjA| = 36$

(D) $A^2 = \begin{bmatrix} -5 & -3 \\ 2 & -6 \end{bmatrix}$

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:

If A is a matrix of order $m × n$ and $B$ is a matrix such that $AB^T$ and $B^TA$ are both well-defined matrices, then order of matrix B is

For a square matrix $A$ of order 3, if $|A| = 2$, then $|adj\ 2A| =$

The linear equation involving $x$ and $y$ are written in matrix form as:

If $A$ is a square matrix of order 3 such that $A(adj\,A)=\begin{bmatrix}-3 & 0 & 0\\0 & -3 & 0\\0 & 0 & -3\end{bmatrix}$, then which of the following is NOT correct?