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

-25

The correct answer is Option (2) → -25

Given: $A$ is a $3\times 3$ matrix and $|A| = 5$.

Use determinant properties:

$|AA^{T}| = |A|\;|A^{T}|$

But $|A^{T}| = |A|$

So:

$|AA^{T}| = 5 \cdot 5 = 25$

$|-AA^{T}| = (-1)^{3}\,|AA^{T}|$ (because multiplying a $3\times 3$ matrix by –1 multiplies determinant by $(-1)^{3}$)

$= -1 \cdot 25$

$= -25$

The value of $|-AA^{T}|$ is $-25$.