Question

If A=`[[1         0],[-1       7]]`, find k such that A² − 8A + kI = 0.

 

Answer 1

\[Given: \hspace{0.167em} A = \begin{bmatrix}1 & 0 \\ - 1 & 7\end{bmatrix}\]

\[\]

\[Now, \]

\[ A^2 = AA\]

\[ \Rightarrow A^2 = \begin{bmatrix}1 & 0 \\ - 1 & 7\end{bmatrix}\begin{bmatrix}1 & 0 \\ - 1 & 7\end{bmatrix}\]

\[ \Rightarrow A^2 = \begin{bmatrix}1 - 0 & 0 + 0 \\ - 1 - 7 & 0 + 49\end{bmatrix}\]

\[ \Rightarrow A^2 = \begin{bmatrix}1 & 0 \\ - 8 & 49\end{bmatrix}\]

 

\[ A^2 - 8A + kI = 0\]

\[ \Rightarrow \begin{bmatrix}1 & 0 \\ - 8 & 49\end{bmatrix} - 8\begin{bmatrix}1 & 0 \\ - 1 & 7\end{bmatrix} + k\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = 0\]

\[ \Rightarrow \begin{bmatrix}1 & 0 \\ - 8 & 49\end{bmatrix} - \begin{bmatrix}8 & 0 \\ - 8 & 56\end{bmatrix} + \begin{bmatrix}k & 0 \\ 0 & k\end{bmatrix} = 0\]

\[ \Rightarrow \begin{bmatrix}1 - 8 + k & 0 - 0 + 0 \\ - 8 + 8 + 0 & 49 - 56 + k\end{bmatrix} = 0\]

\[ \Rightarrow \begin{bmatrix}- 7 + k & 0 \\ 0 & - 7 + k\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]

\[\]

The corresponding elements of two equal matrices are equal . 

\[ \therefore - 7 + k = 0 \]

\[ \Rightarrow k = 7 \]

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