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Question
Show that `[(1, 2),(2, 1)]` is a solution of the matrix equation X² – 2X – 3I = 0,Where I is the unit matrix of order 2
Solution
Given
X2 – 2X – 3I = 0
Solution = `[(1, 2),(2, 1)]`
or
X = `[(1, 2),(2, 1)]`
∴ X2 = `[(1, 2),(2, 1)][(1, 2),(2, 1)]`
= `[(1 + 4, 2 + 2),(2 + 2, 4 + 1)]`
= `[(5, 4),(4, 5)]`
Now X2 - 2X - 3l
= `[(5, 4),(4, 5)] - 2[(1, 2),(2, 1)] - 3[(1, 0),(0, 1)]`
= `[(5, 4),(4, 5)] - [(2, 4),(4, 2)] - [(3, 0),(0, 3)]`
= `[(5 - 2 - 3, 4 - 4 + 0),(4 - 4 - 0, 5 - 2 - 3)]`
= `[(0, 0),(0, 0)]`
∴ X2 = 2X – 31 = 0
Hence proved.
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