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# Which of the following is an example of matrices A,B and C such that AB=AC,where A is non-zero matrix,but $B\neq C.$

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: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
Step1:
Given
AB=AC
$B\neq C.$
A-non zero matrix.
Let $A=\begin{bmatrix}1 & 1\\1 & 1\end{bmatrix} B=\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}C=\begin{bmatrix}3 & 4\\1 & 2\end{bmatrix}$
Step2:
$AB=\begin{bmatrix}1 & 1\\1 & 1\end{bmatrix}\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$
$\;\;=\begin{bmatrix}1+3 & 2+4\\1+3 & 2+4\end{bmatrix}$
$\;\;=\begin{bmatrix}4 & 6\\4 & 6\end{bmatrix}$
Step3:
$AC=\begin{bmatrix}1 & 1\\1 & 1\end{bmatrix}\begin{bmatrix}3 & 4\\1 & 2\end{bmatrix}$
$\;\;\;=\begin{bmatrix}3+1 & 4+2\\3+1 & 4+2\end{bmatrix}$
$\;\;\;=\begin{bmatrix}4 & 6\\4 & 6\end{bmatrix}$
$\Rightarrow AB=AC.$
and $B\neq C$
$A\rightarrow$non zero matrix.
Hence we get AB=AC but A is non zero matrix and and $B\neq C$