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Question
Find `A^2 - 5A + 6I if A = [(2,0,1),(2,1,3),(1,-1,0)]`
Solution
We have A2 = A × A
Given `"A" = [(2,0,1),(2,1,3),(1, -1,0)]`
`"A"^2 = "A". "A" = [(2,0,1),(2,1,3),(1, -1,0)] [(2,0,1),(2,1,3),(1, -1,0)]`
`= [(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)]`
`5A = 5 = [(2, 0, 1),(2, 1, 3),(1, -1,0)] [(10,0,5),(10,5,15),(5, -5,0)]`
`6I = 6 = [(1, 0, 0),(0, 1, 0),(0, 0,1)] [(6,0,0),(0,6,0),(0, 0,0)]`
`"A"^2 - 5 "A" + 6 "I" = [(5, -1, 2), (9, -2, 5),(0, -1,-2)] - [(10,0,5),(10,5,15),(5, -5,0)] + [(6, 0,0),(0,6,0),(0,0,6)]`
`= [(5, -1, 3), (-1, -7, -10),(-5, 4,-2)] - [(6,0,0),(0,6,0),(0,0,6)] + [(1,-1,-3),(-1,-1,-10),(-5,4,4)]`
`= [(1,-1,-3),(-1,-1,-10),(-5, 4, 4)]`
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