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MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES5 Marks
Question
If $\text{A}=\begin{bmatrix}1&5\\7&12\end{bmatrix}$ and $\text{B}=\begin{bmatrix}9&1\\7&8\end{bmatrix},$ find a matrix C such that 3A + 5B + 2C is a null matrix.
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Answer

We have, $\text{A}=\begin{bmatrix}1&5\\7&12\end{bmatrix}$ and $\text{B}=\begin{bmatrix}9&1\\7&8\end{bmatrix}$ Let $\text{C}=\begin{bmatrix}\text{a}&\text{b}\\\text{c}&\text{d}\end{bmatrix}$ $\therefore\ 3\text{A}+5\text{B}+2\text{C}=0$ $\Rightarrow\ \begin{bmatrix}3&15\\21&36\end{bmatrix}+\begin{bmatrix}45&5\\35&40\end{bmatrix}+\begin{bmatrix}2\text{a}&2\text{b}\\2\text{c}&2\text{d}\end{bmatrix}=\begin{bmatrix}0&0\\0&0\end{bmatrix}$ $\Rightarrow\ \begin{bmatrix}48+2\text{a}&20+2\text{b}\\56+2\text{c}&76+2\text{d}\end{bmatrix}=\begin{bmatrix}0&0\\0&0\end{bmatrix}$ $\Rightarrow\ 2\text{a}+48=0\Rightarrow\ \text{a}=-24$ Also, $20+2\text{b}=0\Rightarrow\ \text{b}=-10$ $56+2\text{c}=0\Rightarrow\ \text{c}=-28$And $76+2\text{d}=0\Rightarrow\ \text{d}=-38$
$\therefore\ \text{C}=\begin{bmatrix}-24&-10\\-28&-38\end{bmatrix}$

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