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Question
Find AB, if A = `((1,2,3),(1,-2,-3))` and B = `((1,-1),(1,2),(1,-2))`. Examine whether AB has inverse or not.
Solution
AB = `((1,2,3),(1,-2,-3)) xx ((1,-1),(1,2),(1,-2))`
`= [(1(1) + 2(1) + 3(1),1(-1)+2(2)+3(-2)),(1(1)+(-2)(1)+(-3)(1),1(-1)+(-2)(2)+(-3)(-2))]`
`= [(1+2+3, -1+4-6),(1-2-3 , -1-4+6)]`
`= [(6,-3),(-4,1)]`
∴ |AB| = `|(6,-3),(-4,1)| = 6 - 12 = - 6 ≠ 0`
∴ AB is a non-singular matrix.
Hence, (AB)-1 exists.
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