For the matrix $A =\begin{bmatrix}2&-1&-1\\0&2&3\\1&-2&1\end{bmatrix}$, which of the following statements are correct?

(A) The order of the matrix is 3 × 3
(B) $|A|= 21$
(C) $\text{|adj A| = 225}$
(D) A is skew symmetric matrix

Choose the correct answer from the options given below:

(A), and (C) only

The correct answer is Option (3) → (A), and (C) only

Given: $A = \begin{bmatrix} 2 & -1 & -1 \\ 0 & 2 & 3 \\ 1 & -2 & 1 \end{bmatrix}$

(A) Order of $A$ → 3 rows × 3 columns → Correct

Compute determinant:

$|A| = 2\begin{vmatrix}2 & 3 \\ -2 & 1\end{vmatrix} - (-1)\begin{vmatrix}0 & 3 \\ 1 & 1\end{vmatrix} + (-1)\begin{vmatrix}0 & 2 \\ 1 & -2\end{vmatrix}$

$= 2(2*1 - 3*(-2)) - (-1)(0*1 - 3*1) + (-1)(0*(-2) - 2*1)$

$= 2(2+6) - (-1)(-3) + (-1)(-2) = 2*8 - 3 + 2 = 16 - 3 + 2 = 15$

(B) |A| = 21 → Incorrect

(C) $|\text{adj } A| = |A|^{3-1} = |A|^2 = 15^2 = 225$ → Correct

(D) A skew-symmetric if $A^T = -A$ → Not true → Incorrect