Let $A$ be a $2 \times 2$ matrix, i.e. \begin{equation} A = \begin{pmatrix} X_1 & X_2 \\ X_3 & X_4 \\ \end{pmatrix} \end{equation}

where the $X_{i}$'s entries i.i.d normal random variable with mean $0$ and variance $1$. The problem was to determine the ditribution of $\det(A)$. Now with characteristic function and the law of total expectation I come to the solution, that the distribution function has density $\exp(-|x|)/2$. This was not so difficult.

Now I know that $\det(A)$ is the area of a parallelogram with vertices \begin{align} (0,0), (X_{1},X_{2}), (X_{3},X_{4}), (X_{1} + X_{2},X_{3} + X_{4}). \end{align}

Is there a way to obtain the solution using this geometric way?