Question

If  \[A = \begin{bmatrix}- 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix}\] , find A³.

 

 

Answer 1

\[Here, \] 

\[ A^2 = AA\] 

\[ \Rightarrow A^2 = \begin{bmatrix}- 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix}\begin{bmatrix}- 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix}\] 

\[ \Rightarrow A^2 = \begin{bmatrix}1 + 0 + 0 & 0 + 0 + 0 & 0 + 0 + 0 \\ 0 + 0 + 0 & 0 + 1 + 0 & 0 + 0 + 0 \\ 0 + 0 + 0 & 0 + 0 + 0 & 0 + 0 + 1\end{bmatrix}\] 

\[ \Rightarrow A^2 = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\] 

\[Now, \] 

\[ A^3 = A^2 A\] 

\[ \Rightarrow A^3 = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}- 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix}\] 

\[ \Rightarrow A^3 = \begin{bmatrix}- 1 + 0 + 0 & 0 + 0 + 0 & 0 + 0 + 0 \\ 0 + 0 + 0 & 0 - 1 + 0 & 0 + 0 + 0 \\ 0 + 0 + 0 & 0 + 0 + 0 & 0 + 0 - 1\end{bmatrix}\] 

\[ \Rightarrow A^3 = \begin{bmatrix}- 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix} = A\]