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Question
If `A=([2,0,1],[2,1,3],[1,-1,0])` find A2 - 5A + 4I and hence find a matrix X such that A2 - 5A + 4I + X = 0
Solution
`A=[[2,0,1],[2,1,3],[1,-1,0]]`
`A^2=A A=[[2,0,1],[2,1,3],[1,-1,0]][[2,0,1],[2,1,3],[1,-1,0]]=[[5,-1,2],[9,-2,5],[0,-1,-2]]`
Also,`-5A=[[-10,0,-5],[-10,-5,-15],[-5,5,0]]`
`A^2-5A+4I=[[5,-1,2],[9,-2,5],[0,-1,-2]]-[[-10,0,-5],[-10,-5,-15],[-5,5,0]]+[[4,0,0],[0,4,0],[0,0,4]]=[[-1,-1,-3],[-1,-3,-10],[-5,4,2]]`
Now
`A^2-5A+4I+X=O`
`=>X=-(A^2-5A+4I)`
`=>X=(-1)[[-1,-1,-3],[-1,-3,-10],[-5,4,2]]=[[1,1,3],[1,3,10],[5,-4,-2]]`
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