Question

If A=`[[3,-2],[4,-2]] ` , find k such that A² = kA − 2I₂

 

Answer 1

\[Given: A = \begin{bmatrix}3 & - 2 \\ 4 & - 2\end{bmatrix}\]

\[Now, \]

\[ A^2 = AA\]

\[ \Rightarrow A^2 = \begin{bmatrix}3 & - 2 \\ 4 & - 2\end{bmatrix}\begin{bmatrix}3 & - 2 \\ 4 & - 2\end{bmatrix}\]

\[ \Rightarrow A^2 = \begin{bmatrix}9 - 8 & - 6 + 4 \\ 12 - 8 & - 8 + 4\end{bmatrix}\]

\[ \Rightarrow A^2 = \begin{bmatrix}1 & - 2 \\ 4 & - 4\end{bmatrix}\]

\[\]

\[ A^2 = kA - 2 I_2 \]

\[ \Rightarrow \begin{bmatrix}1 & - 2 \\ 4 & - 4\end{bmatrix} = k\begin{bmatrix}3 & - 2 \\ 4 & - 2\end{bmatrix} - 2\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]

\[ \Rightarrow \begin{bmatrix}1 & - 2 \\ 4 & - 4\end{bmatrix} = \begin{bmatrix}3k & - 2k \\ 4k & - 2k\end{bmatrix} - \begin{bmatrix}2 & 0 \\ 0 & 2\end{bmatrix}\]

\[ \Rightarrow \begin{bmatrix}1 & - 2 \\ 4 & - 4\end{bmatrix} = \begin{bmatrix}3k - 2 & - 2k - 0 \\ 4k - 0 & - 2k - 2\end{bmatrix}\]

\[\]

The corresponding elements of two equal matrices are equal . 

\[ \therefore 1 = 3k - 2 \]

\[ \Rightarrow 1 + 2 = 3k \]

\[ \Rightarrow 3 = 3k \]

\[ \Rightarrow k = 1\]