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)

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 $