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: (B), (D)

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

Assuming matrix to be $A$.
Taking determinant
A)
$|A{|}^2=1$
B)
$|A{|}^2=-1$
C)
$|A{|}^2=1$
D)
​​​​​​​$|A{|}^2=-1$
Square of a determinant cannot be negative.
Hence option B and D are not square of a matrix.