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Question
If A= `((1,0,2),(0,2,1),(2,0,3))` and A3 - 6A2 +7A + kI3 = O find k.
Solution
`A=[(1,0,2),(0,2,1),(2,0,3)]`
`A^2="AA" = [(1,0,2),(0,2,1),(2,0,3)][(1,0,2),(0,2,1),(2,0,3)]`
`=[(5,0,8),(2,4,5),(8,0,13)]`
`A^3=A^2.A=[(5,0,8),(2,4,5),(8,0,13)][(1,0,2),(0,2,1),(2,0,3)]`
`=[(21,0,34),(12,8,23),(34,0,55)]`
∴ A3 - 6A2 + 7A + KI3 = 0
`=>[(21,0,34),(12,8,23),(34,0,55)]-6[(5,0,8),(2,4,5),(8,0,13)]+7[(1,0,2),(0,2,1),(2,0,3)]+k[(1,0,0),(0,1,0),(0,0,1)]=0`
⇒ k = 2
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