Question

If $A=\left[35\right]$ , $B=\left[73\right]$ , then find a non-zero matrix C such that AC=BC

Answer 1

Given two matrices are

$A=\left[35\right]$ , $B=\left[73\right]$

To find out AC and BC, C must be a matrix of order $2\times n$ where n is any natural number

[Number of rows of C must be equal to number of columns of A and B]

Taking $n=1$, Let $C=\left[\begin{array}{c}x\\ y\end{array}\right]$

For $AC=BC$ [ Given ]

$\Rightarrow \left[35\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[73\right]\left[\begin{array}{c}x\\ y\end{array}\right]$

$\Rightarrow [3x+5y]=[7x+3y]$

Equating the corresponding element, we get

$\Rightarrow 3x+5y=7x+3y$

$\Rightarrow 4x=2y$

$\Rightarrow x=\frac{1}{2}y$

$\Rightarrow y=2x$

$\therefore C=\left[\begin{array}{c}x\\ 2x\end{array}\right]$

We can see clearly that on taking C of

order $2\times 1,2\times 2,2\times 3$, … we get

$C=\left[\begin{array}{c}x\\ 2x\end{array}\right],\left[\begin{array}{cc}x& x\\ 2x& 2x\end{array}\right],\left[\begin{array}{ccc}x& x& x\\ 2x& 2x& 2x\end{array}\right]$,……

Hence, in general

$C=\left[\begin{array}{c}k\\ 2k\end{array}\right],\left[\begin{array}{cc}k& k\\ 2k& 2k\end{array}\right]$,etc,

Where k is any real number except zero .