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If A is non-singualr and $A^{2} - 5 A + 7 I = 0$ then I =

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$\frac{1}{7} A - \frac{5}{7} A^{- 1}$

$\frac{1}{7} A + \frac{5}{7} A^{- 1}$

$\frac{1}{5} A + \frac{7}{5} A^{- 1}$

$\frac{1}{5} A - A^{- 1}$

answer is C.

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Detailed Solution

$A^{2} - 5 A + 7 I = 0 A^{- 1} (A^{2} - 5 A + 7 I) = 0 A^{- 1} A^{2} - 5 A^{- 1} A + 7 A^{- 1} I = 0 IA - 5 I + 7 A^{- 1} = 0 5 I = A + 7 A^{- 1} I = \frac{1}{5} A + \frac{7}{5} A^{- 1}$

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