Chapter 11.4, Problem 92AYU

To determine

To explain: If there are three matrices A, B and C such that $AC=BC$ then whether the condition $A=B$ hold or not.

Answer to Problem 92AYU

If there are three matrices A, B and C such that $AC=BC$ then the condition $A=B$ hold if C is an invertible matrix.

Explanation of Solution

Given information:

The property of real numbers $a,b,c\ne 0$ such that if $ac=bc$ then $a=b$ .

Consider three matrices A, B and C such that $AC=BC$ .

Post multiply both sides by $C^{-1}$ .

  $\begin{array}{c}AC{C}^{-1}=BC{C}^{-1}\\ A\left(C{C}^{-1}\right)=B\left(C{C}^{-1}\right)\end{array}$

For a matrix A, the inverse of a $2\times 2$ is given by the formula $A{A}^{-1}=I_2$ , where $I_2$ is the identity matrix.

  $\begin{array}{c}AC{C}^{-1}=BC{C}^{-1}\\ A\left(C{C}^{-1}\right)=B\left(C{C}^{-1}\right)\\ AI=BI\\ A=B\end{array}$

The above expression obtained is only valid if C is an invertible matrix. That is inverse of C exists. C is a nonsingular matrix. Determinant of C is not zero.

Thus, the there are three matrices A, B and C such that $AC=BC$ then the condition $A=B$ hold if C is an invertible matrix.