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If A=[[2 ,3] ,[1, 2]] , verify that A^2-...

If `A=[[2 ,3] ,[1, 2]]` , verify that `A^2-4A+I=O` , where `I=[[1, 0], [0, 1]]` and `O=[[0, 0], [0, 0]]` . Hence, find `A^(-1)` .

Text Solution

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`A=[[2, 3], [1, 2]]`
`A^2=[[2, 3], [1, 2]] [[2, 3], [1, 2]]=[[7, 12], [4, 7]]`
`A^2-4A+I=[[7, 12], [4, 7]]-4[[2, 3], [1, 2]]+[[1,0],[0,1]]=[[0,0],[0,0]]`
Multiplying both sides with `A^(-1)`
`(A^(-1) A)A-4I+A^(-1)=0`
`A^(-1)=4I-A`
`=4[[1,0],[0,1]]-[[2, 3], [1, 2]]`
`=[[2, -3], [-1, 2]]`
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