Prove that matrix
\begin{bmatrix}1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}
can be square of matrix with all real entries.
I have found one such matrix to be
\begin{bmatrix}1&0&0\\0&1&-1\\0&2&-1\end{bmatrix} but is there an elegant way to do it without any trial and error?
Best Answer
Sure. Your matrix is the matrix of a half-turn around the $x$-axis. Just take a quarter of a turn around the same axis:$$\begin{pmatrix}1&0&0\\0&0&-1\\0&1&0\end{pmatrix}\text{ or }\begin{pmatrix}1&0&0\\0&0&1\\0&-1&0\end{pmatrix}.$$