Question

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 ).

# 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

Solution