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If A={bmatrix} 3 & 1 -1 & 2 {bmatrix} show that A-5A+7I=0 hen

Question

If $A = [3 - 1 12]$ show that $A - 5 A + 7 I = 0$ hence find $A^{- 1}$

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Charlotte Discussed

A = [3 - 1 12]

show that

A - 5 A + 7 I = 0

hence find

A^{- 1}

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Text solutionVerified

A^{2} = [3 - 1 12] \times [3 - 1 12] = [8 - 5 53]

5 A = [15 - 5 510], 7 I = [7007]

A^{2} - 5 A + 7 I

= [8 - 5 53] - [15 - 5 510] + [7007] = [0000]

A^{2} - 5 A + 7 I = 0

Pre multiply by

A^{- 1}

A^{- 1} \times A^{2} - 5 A^{- 1} \times A + 7 A^{- 1} \times I = 0

A - 5 I + 7 A^{- 1} = 0

A - 5 I = - 7 A^{- 1}

5 I - A = 7 A^{- 1}

A^{- 1} = \frac{1}{7} (5 I - A)

A^{- 1} = \frac{1}{7} {[5005] - [3 - 1 12]} = \frac{1}{7} [21 - 1 3]

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