Find the real matrices

If $M(a)$ is the matrix $M(a) \; = \; \left(\begin{array}{cc} 1 & a \\ 0 & 1 \end{array}\right)$ then $M(a)^2 = M(2a)$ , and hence $M(a)^2 + \big(M(a)^T\big)^2 \; = \; M(2a) + M(2a)^T \; = \; \left(\begin{array}{cc} 2 & 2a \\ 2a & 2 \end{array}\right)$ so, for this question, we want $M = M(\tfrac32)$ and $N = M^T$ .

If $M,N$ are real matrices such that $M^2 + N^2 = \left(\begin{array}{cc} 2 & 3 \\ 3 & 2 \end{array}\right)$ and $MN=NM$ , then $Z = M+iN$ is a complex matrix such that $Z \,\overline{Z} = \left(\begin{array}{cc}2 & 3 \\ 3 & 2 \end{array}\right)$ , which implies that $\big|\mathrm{det}\,Z\big|^2 \; = \; \mathrm{det}\,Z \times \mathrm{det}\,\overline{Z} \; = \; \mathrm{det}\left(\begin{array}{cc} 2 & 3 \\ 3 & 2 \end{array}\right) \; = \; -5$ which is not possible.