Question

For matrix multiplication, how do I prove that if $AB = AC$, $B$ may not be equal to $C$?

Answer 1

Take different situations for invertible and non-invertible matrices and check the feasibility of left-hand cancellation of $A$.
You can take the help of the identity matrix $I$ to solve the equation and make it more solvable.

Complete step-by-step solution:
This statement is correct for some of the not invertible matrices $A$
If $A$ is an invertible matrix, then${A^{ - 1}}$ exists, and it is such that
A$$$${A^{ - 1}}$$$$ = I$$$$ = {A^{ - 1}}A, where $I$ is the identity matrix.
In this case, $AB = AC$ we could multiply both sides for ${A^{ - 1}}$ to the left, and obtain
${A^{ - 1}}AB = {A^{ - 1}}AC$ which means $B = C$ because A$$$${A^{ - 1}}$$$$ = I.
So, if $A$ is invertible, your statement cannot be proved.
So, $A$ must surely be not invertible i.e. its determinant must be zero. The simplest matrix where every coefficient is zero can be the null matrix.
If you choose it as $A$ you'll obtain that
$AB = AC$ which means $0 = 0$; where $0$ are the zero matrices, regardless of $B$ and $C$
Additional information: An Invertible Matrix is a square matrix defined as invertible if the product of the matrix and its inverse is the identity matrix. An identity matrix is a matrix in which the main diagonal is all $1$ s and the rest of the values in the matrix are$0$s. An invertible matrix is sometimes referred to as nonsingular or non-degenerate and is usually defined using real or complex numbers. The method of detecting a matrix's inverse is called matrix inversion. However, it is important to note that not all matrices are invertible. If a matrix can be multiplied by its inverse then it is said to be invertible. For example, there is no number that can be multiplied by $0$ to get a value of $1$, so the number $0$ has no multiplicative inverse. In addition, the matrix may have a multiplicative inverse (or reciprocal, as in the case of matrices that are not square (different number of rows and columns).

Note: It is important that we know when a matrix is invertible and when it isn’t. The basic idea of invertibility and properties of matrices is a must prerequisite before attempting this question. One must also learn different types of matrices of different orders to ease the question.