Question

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

Answer 1

Hint: 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.