Question-55

If \(\displaystyle A\) is a square matrix of order \(\displaystyle 2\) whose trace is \(\displaystyle 0\), which of the following are true?

• \(\displaystyle A^{2}\) is the zero matrix.
• \(\displaystyle A^{2}\) is the identity matrix.
• \(\displaystyle A^{2}\) is a diagonal matrix.
• \(\displaystyle A^{2}\) is a scalar matrix.

Hint

\[ \begin{equation*} A=\begin{bmatrix} a & b\\ c & -a \end{bmatrix} \end{equation*} \]

Answer

• \(\displaystyle A^{2}\) is the zero matrix.
• \(\displaystyle A^{2}\) is the identity matrix.
• \(\displaystyle A^{2}\) is a diagonal matrix.
• \(\displaystyle A^{2}\) is a scalar matrix.

Solution

We have:

\[ \begin{equation*} A=\begin{bmatrix} a & b\\ c & -a \end{bmatrix} \end{equation*} \]

From this, we see that:

\[ \begin{equation*} A^{2} =\begin{bmatrix} a^{2} +bc & 0\\ 0 & a^{2} +bc \end{bmatrix} =\left( a^{2} +bc\right) I \end{equation*} \]

This is a scalar matrix. Since every scalar matrix is also a diagonal matrix, option-(c) is also correct.