EXAMPLE 3 Let A = 1 1 4 2 2 3 5 2 3...
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EXAMPLE 3 Let A = 1 1 4 2 2 3 5 2 37 and B = 4 5 | -2 1 Find AB and BA, and show that ABBA

11th - 12th Class
Maths
Solution
134
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SOLUTION Here ( A ) is a ( 2 times 3 ) matrix and ( B ) is a ( 3 times 2 ) matrix. So, ( A B ) exists and it is a ( 2 times 2 ) matrix. [ begin{aligned} text { Now, } A B &=left[begin{array}{rrr} 1 & -2 & 3 -4 & 2 & 5 end{array}right]left[begin{array}{rr} 2 & 3 4 & 5 -2 & 1 end{array}right] &=left[begin{array}{rr} 1.2+(-2) cdot 4+3 cdot(-2) & 1 cdot 3+(-2) cdot 5+3 cdot 1 (-4) cdot 2+2 cdot 4+5 cdot(-2) & (-4) cdot 3+2 cdot 5+5 cdot 1 end{array}right] &=left[begin{array}{cc} -12 & -4 -10 & 3 end{array}right] end{aligned} ] Again, ( B ) is a ( 3 times 2 ) matrix and ( A ) is a ( 2 times 3 ) matrix. So, ( B A ) exists and it is a ( 3 times 3 ) matrix. [ begin{aligned} text { Now, } B A &=left[begin{array}{rr} 2 & 3 4 & 5 -2 & 1 end{array}right]left[begin{array}{rrr} 1 & -2 & 3 -4 & 2 & 5 end{array}right] &=left[begin{array}{rrr} 2 & 1+3 cdot(-4) & 2 cdot(-2)+3 cdot 2 & 2 cdot 3+3 cdot 5 4 cdot 1+5 cdot(-4) & 4 cdot(-2)+5 cdot 2 & 4 cdot 3+5 cdot 5 (-2) cdot 1+1 cdot(-4) & (-2) cdot(-2)+1 cdot 2 & (-2) cdot 3+1 cdot 5 end{array}right] &=left[begin{array}{rrr} -10 & 2 & 21 -16 & 2 & 37 -6 & 6 & -1 end{array}right] end{aligned} ] Hence, ( A B eq B A ).
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