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) $|adj\ A| = 225$

(D) A is skew symmetric matrix

Choose the correct answer from the options given below:

For the matrix $A = \begin{bmatrix} 2 & -1 & -1 \\ 0 & 2 & 3 \\ 1 & -2 & 1 \end{bmatrix}$

Statement (A): The order of the matrix is 3 × 3

The matrix has 3 rows and 3 columns.

Order = 3 × 3

Statement (A) is correct.

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Statement (B): |A| = 21

Expanding along the first row:

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

$= 2[(2)(1) - (3)(-2)] + 1[(0)(1) - (3)(1)] - 1[(0)(-2) - (2)(1)]$

$= 2[2 + 6] + 1[0 - 3] - 1[0 - 2]$

$= 2(8) - 3 + 2$

$= 16 - 3 + 2$

$= 15$

Statement (B) is incorrect.

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Statement (C): $|adj\ A| = 225$

For any $n \times n$ matrix: $|adj\ A| = |A|^{n-1}$

With $n = 3$ and $|A| = 15$:

$|adj\ A| = |A|^{3-1}$

$= (15)^2$

$= 225$

Statement (C) is correct.

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Statement (D): A is skew symmetric matrix

A matrix is skew symmetric if $A^T = -A$.

$A^T = \begin{bmatrix} 2 & 0 & 1 \\ -1 & 2 & -2 \\ -1 & 3 & 1 \end{bmatrix}$

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

Since $A^T \neq -A$, the matrix is not skew symmetric.

Statement (D) is incorrect.

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The correct statements are (A) and (C) only.