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

Answer 1

a unit matrix

\[A^2 = AA\] 

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

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

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