Karnataka Board PUCPUC Science 2nd PUC Class 12

If A=[[2,0,1],[2,1,3],[1,-1,0]] , find A2 − 5 A + 16 I. - Mathematics

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If `A=[[2,0,1],[2,1,3],[1,-1,0]]` , find A2 − 5 A + 16 I.

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Solution

`A=[[2,0,1],[2,1,3],[1,-1,0]]`

`therefore A^2=A.A=[[2,0,1],[2,1,3],[1,-1,0]],[[2,0,1],[2,1,3],[1,-1,0]]`

`=>A^2=[[4+0+1,0+0-1,2+0+0],[4+2+3,0+1-3,2+3+0],[2-2+0,0-1+0,1-3+0]]=[[5,-1,2],[9,-2,5],[0,-1,-2]]`

thus

A2 − 5 A + 16 I=`[[5,-1,2],[9,-2,5],[0,-1,-2]]-5[[2,0,1],[2,1,3],[1,-1,0]]+16[[1,0,0],[0,1,0],[0,0,1]]`

`=[[5,-1,2],[9,-2,5],[0,-1,-2]]-[[10,0,5],[10,5,15],[5,-5,0]]+[[16,0,0],[0,16,0],[0,0,16]]`

`=[[5-10+16,-1-0+0,2-5+0],[9-10+0,-2-5+16,5-15+0],[0-5+0,-1+5+0,-2-0+16]]`

`=[[11,-1,-3],[-1,9,-10],[-5,4,14]]`

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