Given |adj A| where A is a square matrix of order two.
We know $|adj (A)|=|A|^{n-1}$
Here n=2.
Therefore $|adj (A)|=|A|^{2-1}$
$\qquad\qquad\qquad\;\;=|A|^1$
$\qquad\qquad\qquad\;\;=|A|$
Therefore $|adj (A)|=|A|$,when A is a square matrix of order two.
Hence |adj A|=$|A|^2$ is False.