1. Form of the matrix

Let

where each entry is either $0$ or $1$.

So there are total

possible matrices, all equally likely.

The determinant is

Since $a,b,c,d\in\{0,1\}$, both $ad$ and $bc$ can only be $0$ or $1$. Hence possible values of $X$ are:

2. Count matrices for each determinant value

We count the number of matrices giving each value.

Case 1: $X=1$

We need

Now $ad=1$ implies

Also $bc=0$ means $(b,c)\neq(1,1)$, so possible pairs are:

Thus number of matrices with determinant $1$ is

Case 2: $X=-1$

We need

Now $bc=1$ implies

Also $ad=0$ means $(a,d)\neq(1,1)$, so possible pairs are:

Thus number of matrices with determinant $-1$ is

Case 3: $X=0$

Remaining matrices:

So number of matrices with determinant $0$ is

Therefore,

3. Find the mean $E[X]$

4. Find $E[X^2]$

Since $X^2=1$ when $X=\pm 1$, and $X^2=0$ when $X=0$,

5. Variance

6. Option check

The variance is

So the correct option is:

D: $\frac{3}{8}$