# Show that

Question:

Show that

(i) $\left[\begin{array}{rr}5 & -1 \\ 6 & 7\end{array}\right]\left[\begin{array}{ll}2 & 1 \\ 3 & 4\end{array}\right] \neq\left[\begin{array}{rr}2 & 1 \\ 3 & 4\end{array}\right]\left[\begin{array}{rr}5 & -1 \\ 6 & 7\end{array}\right]$

(ii) $\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0\end{array}\right]\left[\begin{array}{rrr}-1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4\end{array}\right] \neq\left[\begin{array}{rrr}-1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4\end{array}\right]\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0\end{array}\right]$

Solution:

(i)

$\left[\begin{array}{rr}5 & -1 \\ 6 & 7\end{array}\right]\left[\begin{array}{ll}2 & 1 \\ 3 & 4\end{array}\right]$

$=\left[\begin{array}{ll}5(2)-1(3) & 5(1)-1(4) \\ 6(2)+7(3) & 6(1)+7(4)\end{array}\right]$

$=\left[\begin{array}{ll}10-3 & 5-4 \\ 12+21 & 6+28\end{array}\right]=\left[\begin{array}{ll}7 & 1 \\ 33 & 34\end{array}\right]$

$\left[\begin{array}{rr}2 & 1 \\ 3 & 4\end{array}\right]\left[\begin{array}{rr}5 & -1 \\ 6 & 7\end{array}\right]$

$=\left[\begin{array}{ll}2(5)+1(6) & 2(-1)+1(7) \\ 3(5)+4(6) & 3(-1)+4(7)\end{array}\right]$

$=\left[\begin{array}{ll}10-3 & 5-4 \\ 12+21 & 6+28\end{array}\right]=\left[\begin{array}{ll}7 & 1 \\ 33 & 34\end{array}\right]$

$\therefore\left[\begin{array}{rr}5 & -1 \\ 6 & 7\end{array}\right]\left[\begin{array}{rr}2 & 1 \\ 3 & 4\end{array}\right] \neq\left[\begin{array}{rr}2 & 1 \\ 3 & 4\end{array}\right]\left[\begin{array}{rr}5 & -1 \\ 6 & 7\end{array}\right]$

(ii)

$\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0\end{array}\right]\left[\begin{array}{rrr}-1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4\end{array}\right]$

$=\left[\begin{array}{lll}1(-1)+2(0)+3(2) & 1(1)+2(-1)+3(3) & 1(0)+2(1)+3(4) \\ 0(-1)+1(0)+0(2) & 0(1)+1(-1)+0(3) & 0(0)+1(1)+0(4) \\ 1(-1)+1(0)+0(2) & 1(1)+1(-1)+0(3) & 1(0)+1(1)+0(4)\end{array}\right]$

$=\left[\begin{array}{ccc}5 & 8 & 14 \\ 0 & -1 & 1 \\ -1 & 0 & 1\end{array}\right]$

$\left[\begin{array}{rrr}-1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4\end{array}\right]\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0\end{array}\right]$

$=\left[\begin{array}{llc}-1(1)+1(0)+0(1) & -1(2)+1(1)+0(1) & -1(3)+1(0)+0(0) \\ 0(1)+(-1)(0)+1(1) & 0(2)+(-1)(1)+1(1) & 0(3)+(-1)(0)+1(0) \\ 2(1)+3(0)+4(1) & 2(2)+3(1)+4(1) & 2(3)+3(0)+4(0)\end{array}\right]$

$=\left[\begin{array}{rrr}-1 & -1 & -3 \\ 1 & 0 & 0 \\ 6 & 11 & 6\end{array}\right]$

$\therefore\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0\end{array}\right]\left[\begin{array}{rrr}-1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4\end{array}\right] \neq\left[\begin{array}{rrr}-1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4\end{array}\right]\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0\end{array}\right]$