Let A be any square matrix of order n, then which of the following are true?

(A) $| \text{adj } A| = |A|^{n-1}$

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

(C) $| \text{adj } A| = |A|^n$

(D) $(A^T)^{-1} = (A^{-1})^T$

Choose the correct answer from the options given below:

For a square matrix A of order n, checking each statement:

For statement (A): $|\text{adj } A| = |A|^{n-1}$

The adjoint (or adjugate) matrix of A satisfies the property:

$|\text{adj } A| = |A|^{n-1}$

This is a standard result for square matrices of order n.

Statement (A) is correct.

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For statement (B): $|A^{-1}| = \frac{1}{|A|}$

$A \times A^{-1} = I$

Taking determinant on both sides:

$|A \times A^{-1}| = |I|$

$|A| \times |A^{-1}| = 1$

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

Statement (B) is correct.

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For statement (C): $|\text{adj } A| = |A|^n$

From statement (A), the correct formula is:

$|\text{adj } A| = |A|^{n-1}$

The power is $(n-1)$, not $n$.

Statement (C) is incorrect.

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For statement (D): $(A^T)^{-1} = (A^{-1})^T$

$A \times A^{-1} = I$

Taking transpose of both sides:

$(A \times A^{-1})^T = I^T$

$(A^{-1})^T \times A^T = I$

This shows that $(A^{-1})^T$ is the inverse of $A^T$.

$(A^T)^{-1} = (A^{-1})^T$

Statement (D) is correct.

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The correct statements are (A), (B), and (D).

The incorrect statement is (C).