If $A$ is a square matrix of order 3 and $|A| = 5$, then the value of $|-AA^T|$ is

Given $A$ is a 3×3 square matrix (order 3) and $|A| = 5$.

The expression $|-AA^T|$ represents the determinant of $(-A)$ multiplied by the transpose of matrix $A$.

$|-AA^T| = |(-A) \times A^T|$

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When two matrices multiply, their determinants also multiply:

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

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When a matrix is multiplied by $-1$, the determinant is affected by $(-1)^n$, where $n$ is the order of the matrix.

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

Since the matrix has order 3:

$|{-A}| = (-1)^3 \times 5$

$|{-A}| = -1 \times 5$

$|{-A}| = -5$

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The determinant of a matrix equals the determinant of its transpose:

$|A^T| = |A| = 5$

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$|-AA^T| = |{-A}| \times |A^T|$

$|-AA^T| = (-5) \times 5$

$|-AA^T| = -25$

Therefore, the value of $|-AA^T| = -25$.