Chapter 8.2, Problem 81E

To determine

To verify:The right cancellation law in matrix multiplication.

Answer to Problem 81E

Left cancellation law holds true for real numbers under multiplication, where it is not necessarily holds true for matrices even though the entries are real numbers.

Explanation of Solution

Given information:

The given matrices are

  $A=\left[\begin{array}{cc}0& 1\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right],C=\left[\begin{array}{cc}2& 3\\ 2& 3\end{array}\right]$

Calculations:

  $\begin{array}{l}AC=BC\\ \Rightarrow \left[\begin{array}{cc}0& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}2& 3\\ 2& 3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right]\left[\begin{array}{cc}2& 3\\ 2& 3\end{array}\right]\\ \Rightarrow \left[\begin{array}{cc}0\times 2+1\times 2& 0\times 3+1\times 3\\ 0\times 2+1\times 2& 0\times 3+1\times 3\end{array}\right]=\left[\begin{array}{cc}1\times 2+0\times 2& 1\times 3+0\times 3\\ 1\times 2+0\times 2& 1\times 3+0\times 3\end{array}\right]\\ \Rightarrow \left[\begin{array}{cc}2& 3\\ 2& 3\end{array}\right]=\left[\begin{array}{cc}2& 3\\ 2& 3\end{array}\right]\end{array}$

Conclusion:

Though $AC=BC$ , the matrices $A$ and $B$ are not necessarily equal to each other.