Why the restriction for u?

For convenient purposes, let $B = adj(A)$

$\Rightarrow \det(adj(B)) = |B|^2 \Rightarrow 23^4 = |B|^2 \Rightarrow |B| = \pm 23^2$

$|B| = |adj(A)| = |A|^2$

Now since all entries are real, $|B| \neq -23^2$

$\Rightarrow |adj(A)| = 23^2 \Rightarrow |A|^2 = 23^2 \Rightarrow |A| = \pm 23$

Case 1: $|A| = 23$

$|A| = 3u + 11 = 23 \Rightarrow u = \boxed{4}$

Case 2: $|A| = -23$

$|A| = 3u + 11 = -23 \Rightarrow u = \boxed{-11.\overline{3}}$

Hence the integral value of $u$ is $\boxed{4}$ .

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Note:

If $u = -\dfrac{11}{3}$ , then $|A| = 0$ , which is not possible, since the adjoint of A exists and is non-zero.