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

(A) A is singular matrix

(B) |3A| = 135

(C) |adj A| = 125

(D) $|A^{-1}| = \frac{1}{5}$

Choose the correct answer from the options given below:

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

Finding the determinant of $A$ using first row expansion:

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

$|A| = 2(1 \times 1 - 2 \times 0) + 1(1 \times 1 - 2 \times (-1)) + 0$

$|A| = 2(1) + 1(1 + 2)$

$|A| = 2 + 3$

$|A| = 5$

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Checking Statement (A): A is singular matrix

A matrix is singular when its determinant equals zero.

Since $|A| = 5 \neq 0$, matrix $A$ is not singular.

Statement (A) is false.

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Checking Statement (B): $|3A| = 135$

For an $n \times n$ matrix, $|kA| = k^n \times |A|$

For the $3 \times 3$ matrix with $k = 3$:

$|3A| = 3^3 \times |A|$

$|3A| = 27 \times 5$

$|3A| = 135$

Statement (B) is true.

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Checking Statement (C): $|adj A| = 125$

For an $n \times n$ matrix, $|adj A| = |A|^{n-1}$

For the $3 \times 3$ matrix:

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

$|adj A| = |A|^2$

$|adj A| = 5^2$

$|adj A| = 25$

Statement (C) is false.

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Checking Statement (D): $|A^{-1}| = \frac{1}{5}$

For any invertible matrix, $|A^{-1}| = \frac{1}{|A|}$

$|A^{-1}| = \frac{1}{5}$

Statement (D) is true.

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Therefore, statements (B) and (D) are correct.