If A is a non-singular matrix of order 3 such that $\text{|adj(A)| = 121}$, then $|AA^T|$ is equal to:

121

The correct answer is Option (1) → 121

Given: $|adj(A)| = 121$, $A$ is non-singular of order 3

Property: For a square matrix of order $n$, $|adj(A)| = |A|^{n-1}$

Here, $n = 3 \Rightarrow |adj(A)| = |A|^{3-1} = |A|^2$

$|A|^2 = 121 \Rightarrow |A| = \pm 11$

Since $A$ is non-singular, $|A| \neq 0$, so $|A| = 11$

Now, $|A A^T| = |A| \cdot |A^T| = |A| \cdot |A| = |A|^2 = 11^2 = 121$

$|A A^T| = 121$