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Questions
If A = `[(3, 1),(-1, 2)]`, prove that A2 – 5A + 7I = 0, where I is unit matrix of order 2
If A = `[(3, 1),(-1, 2)]`, prove that A2 – 5A + 7I = 0, where I is 2 × 2 unit matrix and 0 is zero matrix of order 2.
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Solution
Given: A = `[(3, 1),(-1, 2)]`
A2 = A · A
= `[(3, 1),(-1, 2)] [(3, 1),(-1, 2)]`
= `[(9 - 1, 3 + 2),(-3 - 2, -1 + 4)]`
A2 = `[(8, 5),(-5, 3)]`
Now consider
L.H.S. = A2 – 5A + 7I
= A . A – 5A + 7I
= `[(8, 5),(-5, 3)] -5[(3, 1),(-1, 2)] + 7[(1, 0),(0, 1)]`
= `[(8, 5),(-5, 3)] - [(15, 5),(-5, 10)] + [(7, 0),(0, 7)]`
= `[(8 - 15 + 7, 5 - 5 + 0),(-5 + 5 + 0, 3 - 10 + 7)]`
= `[(-7,0),(0,-7)] + [(7,0),(0,7)]`
= `[(0, 0),(0, 0)]`
= 0 = R.H.S.
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