Chapter 4: Q30. (page 172)

Use $A = [\begin{matrix} 1 & - 2 \\ 4 & 3 \end{matrix}]$ , $B = [\begin{matrix} - 5 & 2 \\ 4 & 3 \end{matrix}]$ , and $C = [\begin{matrix} 5 & 1 \\ 2 & - 4 \end{matrix}]$ and scalar $c = 3$ to determine whether the equation is true for the given matrices.
$A B C = C B A$

Short Answer

Expert verified

The provided equation is false.

Step by step solution

Step 1- Determine the value of ABC

Substitute the provided matrices into ABC.

$A B C = [\begin{matrix} 1 & - 2 \\ 4 & 3 \end{matrix}] \cdot [\begin{matrix} - 5 & 2 \\ 4 & 3 \end{matrix}] \cdot [\begin{matrix} 5 & 1 \\ 2 & - 4 \end{matrix}]$

First perform the multiplication between the matrices A and B after it multiply the resulting matrix with C.

$\begin{matrix} A B C = [\begin{matrix} - 13 & - 4 \\ - 8 & 17 \end{matrix}] \cdot [\begin{matrix} 5 & 1 \\ 2 & - 4 \end{matrix}] \\ = [\begin{matrix} - 73 & 3 \\ - 6 & - 76 \end{matrix}] \end{matrix}$

– Determine the value of CBA

Substitute the provided matrices into CBA.

$C B A = [\begin{matrix} 5 & 1 \\ 2 & - 4 \end{matrix}] \cdot [\begin{matrix} - 5 & 2 \\ 4 & 3 \end{matrix}] \cdot [\begin{matrix} 1 & - 2 \\ 4 & 3 \end{matrix}]$

First perform the multiplication between the matrices C and B after it multiply the resulting matrix with A.

$\begin{matrix} C B A = [\begin{matrix} - 21 & 13 \\ - 26 & - 8 \end{matrix}] \cdot [\begin{matrix} 1 & - 2 \\ 4 & 3 \end{matrix}] \\ = [\begin{matrix} 31 & 81 \\ - 58 & 28 \end{matrix}] \end{matrix}$