$\DeclareMathOperator\Prob{Prob}$Let $X,Y\in M_n(\mathbb{R})$ be $2$ random matrices. The entries of $X,Y$ are i.i.d. variables. They follow the standard normal law $N(0,1)$.

i) When $n=2,3,4$, one can show that the expectation of $\det(X^2+Y^2)$ is $(n+1)!$.

Computational tests seem to confirm these results for $n\geq5$.

Is the following true ?

Conjecture 1. For every $n\geq 5$, $E(\det(X^2+Y^2))=(n+1)!$.

ii) If $X_0,Y_0$ is the result of an experiment, then let $p_n=\Prob(\det({X_0}^2+{Y_0}^2)<0)$.

Some experiments give the following approximations.

$n=2,p_2\approx 0.1178;n=4,p_4\approx 0.327;n=6,p_6\approx 0.438; n=8,p_8\approx 0.480$; $n=10,p_{10}\approx 0.494;n=20,p_{20}\approx 0.49997;n=40,p_{40}\approx 0.49999$.

Is the following true ?

Conjecture 2. The sequence $(p_n)$ is increasing and tends to $\dfrac{1}{2}$ when $n$ tends to $+\infty$.

If yes and $n$ is great, then $$\Prob(\det({X_0}^2+{Y_0}^2)<0)\approx \Prob(\det({X_0}^2+{Y_0}^2)>0).$$

EDIT. Thanks to everyone who gave a nice answer to conjecture 1.

I don't know if conjecture 1 helps prove conjecture 2.

The problem is that the positive values taken by the determinant are much larger - in absolute value - than the negative ones.

$\bullet$ Note that, since $X$ is "always" invertible

$\Prob(\det(X^2+Y^2)<0)=\Prob(\det(I+X^{-2}Y^2)<0)=\Prob(\det(I+(YX^{-1})(X^{-1}Y))<0)$.

$\bullet$ With the same hypothesis as above concerning the $(X_i)$, it seems that for every $p$ and for $n$ great enough,

$\Prob(\det({X_1}^2+\cdots+{X_p}^2)<0)\approx 1/2$. ​

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