Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES4 Marks
Question
If $\text{A}=\begin{bmatrix}3&-2\\4&-2\end{bmatrix},$ find k such that A2 = kA - 2I2.
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Answer
Given: $\text{A}=\begin{bmatrix}3&-2\\4&-2\end{bmatrix}$ Now, $\text{A}^2=\text{AA}$ $\Rightarrow\text{A}^2=\begin{bmatrix}3&-2\\4&-2\end{bmatrix}\begin{bmatrix}3&-2\\4&-2\end{bmatrix}$ $\Rightarrow\text{A}^2=\begin{bmatrix}9-8&-6+4\\12-8&-8+4\end{bmatrix}$ $\Rightarrow\text{A}^2=\begin{bmatrix}1&-2\\4&-4\end{bmatrix}$ $\text{A}^2=\text{kA}-2\text{I}_2$ $\Rightarrow\begin{bmatrix}1&-2\\4&-4\end{bmatrix}=\text{k}\begin{bmatrix}3&-2\\4&-2\end{bmatrix}-2\begin{bmatrix}1&0\\0&1\end{bmatrix}$ $\Rightarrow\begin{bmatrix}1&-2\\4&-4\end{bmatrix}=\begin{bmatrix}3\text{k}&-2\text{k}\\4\text{k}&-2\text{k}\end{bmatrix}-\begin{bmatrix}2&0\\0&2\end{bmatrix}$ $\Rightarrow\begin{bmatrix}1&-2\\4&-4\end{bmatrix}=\begin{bmatrix}3\text{k}-2&-2\text{k}-0\\4\text{k}-0&-2\text{k}-2\end{bmatrix}$ The corresponding elements of two equal matrices are equal. $\therefore\ 1=3\text{k}-2$ $\Rightarrow1+2=3\text{k}$ $\Rightarrow3=3\text{k}$ $\Rightarrow\text{k}=1$
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