Question

If \[A = \begin{bmatrix}1 & - 1 \\ - 1 & 1\end{bmatrix}\], satisfies the matrix equation A² = kA, write the value of k.

 

Answer 1

\[A^2 = AA\] 

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

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

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

\[Now, \] 

\[ A^2 = kA\] 

\[ \Rightarrow \begin{bmatrix}2 & - 2 \\ - 2 & 2\end{bmatrix} = k\begin{bmatrix}1 & - 1 \\ - 1 & 1\end{bmatrix}\] 

\[ \Rightarrow \begin{bmatrix}2 & - 2 \\ - 2 & 2\end{bmatrix} = \begin{bmatrix}k & - k \\ - k & k\end{bmatrix}\] 

\[ \therefore k = 2\]