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Why does cancellation law not hold true for matrices I.e if A,B and C are three matrices of orders that are suitable for multiplication such that A.B=A.C Then even if A≠0 still B may or may not be equal to C

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    Unlike real numers, `A≠0` is not a sufficient condition for matrix $A$ to have a multiplicative inverse, which is what the cancellation law relies upon.. – 2017-02-12

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Here's an example: Say $A=\begin{pmatrix}1& 0 \\0&0\end{pmatrix},$ $B= \begin{pmatrix} 1 \\ 1\end{pmatrix}$ and $C= \begin{pmatrix} 1 \\ 2\end{pmatrix}.$ Then $AB= AC = \begin{pmatrix} 1 \\ 0\end{pmatrix}$ but $B\ne C.$ The key is that $A$ is not an invertible matrix. Cancellation only holds when $A^{-1}$ exists.

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    Okay...I got your point.thnx – 2017-02-12
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You can consider a matrix is a linear transformation. If two vectors after transformed by a transformation can be the same vector. Then the cancellation law doesn't hold in matrix