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MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES5 Marks
Question
If $\text{A}=\begin{bmatrix}3&5\end{bmatrix}$ and $\text{B}=\begin{bmatrix}7&3\end{bmatrix},$ then find a non-zero matrix C such that AC = BC.
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Answer

We have, $\text{A}=\begin{bmatrix}3&5\end{bmatrix}_{2\times1}$ and $\text{B}=\begin{bmatrix}7&3\end{bmatrix}_{1\times2}$ Let $\text{C}=\begin{bmatrix}\text{x}\\\text{y}\end{bmatrix}_{2\times1}$ is a non-zero matrix of order 2 × 1. $\therefore\ \text{AC}=\begin{bmatrix}3&5\end{bmatrix}\begin{bmatrix}\text{x}\\\text{y}\end{bmatrix}=\begin{bmatrix}3\text{x}+5\text{y}\end{bmatrix}$ And $\text{BC}=\begin{bmatrix}7&3\end{bmatrix}\begin{bmatrix}\text{x}\\\text{y}\end{bmatrix}=\begin{bmatrix}7\text{x}+3\text{y}\end{bmatrix}$ For AC = BC,$\begin{bmatrix}3\text{x}+5\text{y}\end{bmatrix}=\begin{bmatrix}7\text{x}+3\text{y}\end{bmatrix}$
On using equality of matrix, we get $\Rightarrow\ 3\text{x}+5\text{y}=7\text{x}+3\text{y}$ $\Rightarrow\ 4\text{x}=2\text{y}$ $\Rightarrow\ \text{x}=\frac{1}{2}\text{y}$ $\Rightarrow\ \text{y}=2\text{x}$ $\therefore\ \text{C}=\begin{bmatrix}\text{x}\\2\text{x}\end{bmatrix}$ We see that on taking C of order 2 × 1, 2 × 2, 2 × 3, .....we get $\text{C}=\begin{bmatrix}\text{x}\\2\text{x}\end{bmatrix},\begin{bmatrix}\text{x}&\text{x}\\2\text{x}&2\text{x}\end{bmatrix},\begin{bmatrix}\text{x}&\text{x}&\text{x}\\2\text{x}&2\text{x}&2\text{x}\end{bmatrix}....$ In general, $\text{C}=\begin{bmatrix}\text{k}\\2\text{k}\end{bmatrix},\begin{bmatrix}\text{k}&\text{k}\\2\text{k}&2\text{k}\end{bmatrix}\ \text{etc }.....$Where, k is any real number.

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