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If A=((1,0,2),(0,2,1),(2,0,3)) and A3 - 6A2 +7A + kI3 = O find k. - Mathematics

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If A= `((1,0,2),(0,2,1),(2,0,3))` and A3 - 6A2 +7A + kI3 = O find k.

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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

Concept: Introduction of Operations on Matrices
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