Question

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

• A. $\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}$
• B. $\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&-1\end{bmatrix}$
• C. $\begin{bmatrix}1&0&0\\ 0&-1&0\\ 0&0&-1\end{bmatrix}$
• D. $\begin{bmatrix}-1&0&0\\ 0&-1&0\\ 0&0&-1\end{bmatrix}$

Answer 1

Answer: (B), (D)

$A^{2}=B$
$| A |=\pm \sqrt{|B|}$
$\therefore | B | \neq-1$