If A = [ rrr 1 & -2 & 3 -4 & 2 & 5 ] and B= [ ll 2 & 3 4 & 5 2 & … - Vidyadip

Question

If A = $\left[\begin{array}{rrr} {1} & {-2} & {3} \\ {-4} & {2} & {5} \end{array}\right] \text { and } B=\left[\begin{array}{ll} {2} & {3} \\ {4} & {5} \\ {2} & {1} \end{array}\right]$ , then find AB, BA. Show that AB $\neq$ BA

✓

Answer

Since A is a 2 $\times$ 3 matrix and B is 3 $\times$ 2 matrix.
Hence AB and BA are both defined and are matrices of order 2 $\times$ 2 and 3 $\times$ 3, respectively.
Now,$A B=\left[\begin{array}{rrr} {1} & {-2} & {3} \\ {-4} & {2} & {5} \end{array}\right]\left[\begin{array}{ll} {2} & {3} \\ {4} & {5} \\ {2} & {1} \end{array}\right]$ = $\left[\begin{array}{cc} {2-8+6} & {3-10+3} \\ {-8+8+10} & {-12+10+5} \end{array}\right]$ = $\left[\begin{array}{rr} {0} & {-4} \\ {10} & {3} \end{array}\right]$
and $B A=\left[\begin{array}{cc} {2} & {3} \\ {4} & {5} \\ {2} & {1} \end{array}\right]\left[\begin{array}{ccc} {1} & {-2} & {3} \\ {-4} & {2} & {5} \end{array}\right]$ = $\left[\begin{array}{ccc} {2-12} & {-4+6} & {6+15} \\ {4-20} & {-8+10} & {12+25} \\ {2-4} & {-4+2} & {6+5} \end{array}\right]$ = $\left[\begin{array}{ccc} {-10} & {2} & {21} \\ {-16} & {2} & {37} \\ {-2} & {-2} & {11} \end{array}\right]$
Clearly AB $\neq$ BA
In the above example, both AB and BA are of different orders and so AB $\neq$ BA.

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