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Question
Show that AB ≠ BA in each of the following cases:
`A = [[1,3,-1],[2,-1,-1],[3,0,-1]]` And `B= [[-2,3,-1],[-1,2,-1],[-6,9,-4]]`
Solution
`AB = [[1 3 -1],[2 -1 -1],[3 0 -1]]`` [[-2 3 -1],[-1 2 -1],[-6 9 -4]]`
`⇒ AB = [[-2-3+ 6 3+6-9 -1-3+4],[-4+1+6 6-2-9 -2+1+4],[-6-0+6 9+0-9 -3-0+4]]`
`⇒AB=[[1 0 0],[3 -5 3],[0 0 1]]`..........................(1)
Also,
`BA=`` [[-2 3 -1],[-1 2 -1],[-6 9 -4]]``[[1 3 -1],[2 -1 -1],[3 0 -1]]`
`⇒BA=[[-2+6-3 -6-3+0 2-3+1],[-1+4-3 -3-2+0 1-2+1],[-6+18-12 -18-9+0 6-9+4]]`
`⇒BA=[[1 -9 0],[0 -5 0],[0 -27 1]]` ...(2)
∴ AB ≠ BA (From eqs. (1) and (2))
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