Question

another important quality of matrix algebra is that it does not have a general cancellation property for matrix multiplication. that is, when ( ac = bc ), it is not necessarily true that ( a = b ).
example 5: an example in which cancellation is not valid
show that ( ac = bc ).
( a=left\begin{array}{ll}1 & 3 \\ 0 & 1end{array}
ight, b=left\begin{array}{ll}2 & 4 \\ 2 & 3end{array}
ight, c=left\begin{array}{cc}1 & -2 \\ -1 & 2end{array}
ight )
solution
( ac=left\begin{array}{ll}1 & 3 \\ 0 & 1end{array}
ightleft\begin{array}{cc}1 & -2 \\ -1 & 2end{array}
ight=left\begin{array}{cc}-2 & 4 \\ -1 & 2end{array}
ight, bc=left\begin{array}{ll}2 & 4 \\ 2 & 3end{array}
ightleft\begin{array}{cc}1 & -2 \\ -1 & 2end{array}
ight=left\begin{array}{cc}-2 & 4 \\ -1 & 2end{array}
ight )
( ac = bc ), even though ( a
eq b ).

Explanation:

Step1: Matrix multiplication rule

For two matrices \(M = [m_{ij}]\) of size \(m\times n\) and \(N=[n_{ij}]\) of size \(n\times p\), the product \(MN = [p_{ij}]\) where \(p_{ij}=\sum_{k = 1}^{n}m_{ik}n_{kj}\)

Step2: Calculate \(AC\)

$$
LATEXBLOCK0
$$

Step3: Calculate \(BC\)

$$
LATEXBLOCK1
$$

Answer:

Since \(AC=

$$\begin{bmatrix}-2&4\\-1&2\end{bmatrix}$$

\) and \(BC=

$$\begin{bmatrix}-2&4\\-1&2\end{bmatrix}$$

\), we have shown that \(AC = BC\) while \(A=

$$\begin{bmatrix}1&3\\0&1\end{bmatrix}$$

eq

$$\begin{bmatrix}2&4\\2&3\end{bmatrix}$$

=B\)