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Question
If \[A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & - 1\end{bmatrix}\] , then A2 is equal to ___________ .
Options
a null matrix
a unit matrix
−A
A
Solution
\[A^2 = AA\]
\[ \Rightarrow A^2 = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & - 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & - 1\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}1 + 0 + 0 & 0 + 0 + 0 & 0 + 0 - 0 \\ 0 + 0 + 0 & 0 + 1 + 0 & 0 + 0 - 0 \\ a + 0 - a & 0 + b - b & 0 + 0 + 1\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\]
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