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# If A=[3 , 5],B=[7, 3],then find a non-zero matrix C such that AC=BC.

Toolbox:
• If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
• If the order of 2 matrices are equal, their corresponding elements are equal, i.e, if $A_{ij}=B_{ij}$, then any element $a_{ij}$ in matrix A is equal to corresponding element $b_{ij}$ in matrix B.
Step1:
Given A=[3,5]$_{1\times 2}$
B=[7,3]$_{1\times 2}$
$\Rightarrow C$ will a be of order $2\times 1$
Let $C=\begin{bmatrix}P\\Q\end{bmatrix}$
Step2:
Given AC=BC.
$[3\;\; 5]\begin{bmatrix}P\\Q\end{bmatrix}=[7 \;\;3]\begin{bmatrix}P\\Q\end{bmatrix}$
3P+5Q=7P+3Q
3P-7P=-5Q+3Q
-4P=-2Q.
P/Q=-2/-4
P/Q=1/2
Q=2P
Hence matrix C can be equal to
$\begin{bmatrix}k \\2k\end{bmatrix}\begin{bmatrix}2k \\4k\end{bmatrix}$.........etc