Question

If $A = \begin{bmatrix}i&0\\\ 0&-i\end{bmatrix}, B = \begin{bmatrix}0&-1\\\ 1&0\end{bmatrix}$ and $C= \begin{bmatrix}0&i\\\ i&0\end{bmatrix}$ then $A^{2} = B^{2} =C^{2}$ is equal to :

• A. $I^2$
• B. I
• C. - I
• D. 2I

Answer 1

Answer: (C)

- I

Answer 2

Let $A = \begin{bmatrix}i&0\\\ 0&-i\end{bmatrix}, B = \begin{bmatrix}0&-1\\\ 1&0\end{bmatrix}$ and $C= \begin{bmatrix}0&i\\\ i&0\end{bmatrix}$ $A. A= \begin{pmatrix}i&0\\\ 0&-i\end{pmatrix} \begin{pmatrix}i&0\\\ 0&-1\end{pmatrix}$ $= \begin{pmatrix}i^{2}&0\\\ 0&i^{2}\end{pmatrix} = \begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix} = - I$ and $B.B = \begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}$ $= \begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix} = -I$ $C.C = \begin{pmatrix}0&i\\\ i&0\end{pmatrix}\begin{pmatrix}0&i\\\ i&0\end{pmatrix} = \begin{pmatrix}i^{2}&0\\\ 0&i^{2}\end{pmatrix} = -I$ Hence, $A^{2} = B^{2} = C^{2} = - I$