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If A= [(1,1,3),(5,2,6),(-2,-1,-3)] then ...

If `A= [(1,1,3),(5,2,6),(-2,-1,-3)]` then find `A^(14)+3A-2I`

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Let `A=[(1,1,3),(5,2,6),(-2,-1,-3)]`
`A^(2)=AxxA=[(1,1,3),(5,2,6),(-2,-1,-3)]xx[(1,1,3),(5,2,6),(-2,1,-3)]`
`=[(1+5-6,1+2-3,3+6-9),(5+10-12,5+4-6,15+12-18),(-2-5+6,-2-2+3,-6-6+9)]`
`=[(0,0,0),(3,3,9),(-1,-1,-3)]`
`:. A^(3)=A^(2)xxA=[(0,0,0),(3,3,9),(-1,-1,-3)]xx[(1,1,3),(5,2,6),(-2,-1,-3)]`
`=[(0+0+0,0+0+0,0+0+0),(3+15-18,3+6-9,9+18-27),(-1-5+6,-1-2+3,9+18-27)]`
`=[(0,0,0),(0,0,0),(0,0,0)]=O`
`:. A^(3)=O`
Hence, A is nilpotent of order 3.
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