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If A+I={:[(2,2,3),(3,-1,1),(4,2,2)]:} th...

If `A+I={:[(2,2,3),(3,-1,1),(4,2,2)]:}` then show that `A^(3)-23A-40I=0`

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`A+l={:[(2,2,3),(3,-2,1),(4,2,2)]:}`
`:.A={:[(2,2,3),(3,-2,1),(4,2,2)]:}-{:[(1,0,0),(0,1,0),(0,0,1)]:}={:[(1,2,3),(3,-2,1),(4,2,1)]:}`
Now, `A^(2)=A A={:[(1,2,3),(3,-2,1),(4,2,1)]:}{:[(1,2,3),(3,-2,1),(4,2,1)]:}={:[(19,4,8),(1,12,8),(14,6,16)]:}`
`A^(3)=A^(2)A={:[(1,2,3),(3,-2,1),(4,2,1)]:}{:[(19,4,8),(1,12,8),(14,6,15)]:}={:[(63,46,69),(69,-6,23),(92,46,63)]:}`
Now, `A^(3)-23A-40={:[(63,46,69),(69,-6,23),(92,46,63)]:}-23{:[(1,2,3),(3,-2,1),(4,2,1)]:}-40{:[(1,0,0),(0,1,0),(0,0,1)]:}`
`={:[(63,46,69),(69,-6,23),(92,46,63)]:}+{:[(-23,-46,-69),(-69,46,-23),(-92,-46,-23)]:}+{:[(-40,0,0),(0,-40,0),(0,0,-40)]:}`
`={:[(0,0,0),(0,0,0),(0,0,0)]:}=0`
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