Let [A = ( pmatrix 0 & 0 & - 1 0 & - 1 & 0 - 1 & 0 & 0 pmatrix )].

Question

Let $$A = \left( {\begin{pmatrix} 0 & 0 & { - 1}  \
   0 & { - 1} & 0  \
   { - 1} & 0 & 0  \ \end{pmatrix} } \right)$$. The only correct
statement about the matrix $A$ is

Accepted Answer

$$A = \left[ {\begin{pmatrix} 0 & 0 & { - 1}  \
   0 & { - 1} & 0  \
   { - 1} & 0 & 0  \

\end{pmatrix} } \right]$$

clearly $$\,\,\,A \ne 0.\,$$ Also $$\,\,\left| A \right| =  - 1 \ne 0$$

$$\therefore$$ $${A^{ - 1}}\,\,$$ exists, further

$$\left( { - 1} \right)I = \left[ {\begin{pmatrix} { - 1} & 0 & 0  \
   0 & { - 1} & 0  \
   0 & 0 & { - 1}  \

\end{pmatrix} } \right] \ne A$$

Also $${A^2} = \left[ {\begin{pmatrix} 0 & 0 & { - 1}  \
   0 & { - 1} & 0  \
   { - 1} & 0 & 0  \

\end{pmatrix} } \right]\left[ {\begin{pmatrix} 0 & 0 & { - 1}  \
   0 & { - 1} & 0  \
   { - 1} & 0 & 0  \

\end{pmatrix} } \right]$$

$$ = \left[ {\begin{pmatrix} 1 & 0 & 0  \
   0 & 1 & 0  \
   0 & 0 & 1  \

\end{pmatrix} } \right] = I$$

Solutions · 1