Question

Which of the following is (are) NOT the square of a $3\times 3$ matrix with real entries?

• A. $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$
• B. $\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]$
• C. $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
• D. $\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]$

Answer 1

Answer: (A), (D)

The correct options are
A $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$
D $\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]$

For $A$ and $D$

$det\left(A^2\right)=-1$

$\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right|=-1=\left|\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right|$

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& -2\\ 0& 1& -1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& -2\\ 0& 1& -1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]$

For C

$I^2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Hence, A and D are NOT the square of a $3\times 3$ matrix with real entries.