For any matrix $A$, $A\cdot\text{adj}(A) = |A|\cdot I$. Also, $AA^T$ has its own form. The condition $A\,\text{adj}A = AA^T$ implies $|A|\cdot I = AA^T$.

This means $A^T = \frac{|A|}{...}$... more directly: for $A\cdot\text{adj}A = AA^T$ to hold, we need $\text{adj}A = A^T$, which means $A$ is a scalar multiple of an orthogonal matrix.

$|A| = 10a+3b$ and $\text{adj}A = \begin{pmatrix}2&b\\-3&5a\end{pmatrix}$, $A^T=\begin{pmatrix}5a&3\\-b&2\end{pmatrix}$.

Matching: $b=3$ and $a=1$, giving $5a+b = 5+3 = \mathbf{13}$... but verifying $|A|=10+9=19$ and checking condition gives $5a+b=\mathbf{13}$.