If A is an invertible matrix, then which of the following statement(s) is/are TRUE?

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

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

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

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

Choose the correct answer from the options given below:

For an invertible matrix $A$, checking each statement:

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

Taking determinant of both sides:

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

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

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

Since $|A^{-1}| = \frac{1}{|A|}$ and not $|A|$, statement (A) is FALSE.

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If $A^{-1}$ is the inverse of $A$, then $A$ is the inverse of $A^{-1}$.

By definition of inverse:

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

Statement (B) is TRUE.

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For an invertible matrix, the inverse formula is:

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

This is a standard property of invertible matrices.

Statement (C) is TRUE.

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Starting with $A \times A^{-1} = I$, taking transpose of both sides:

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

Using $(AB)^T = B^T A^T$:

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

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

Therefore:

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

Statement (D) is TRUE.

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Statements (B), (C), and (D) are TRUE.

Statement (A) is FALSE.