First slide

Applications of determinant

Question

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

Easy

A

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

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

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

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

Solution

$A^{2} - 5 A + 7 I = 0$
By right multiplication with $A^{- 1}$
$A A A^{- 1} - 5 A A^{- 1} + 7 I A^{- 1} = 0 A - 5 I + 7 A^{- 1} = 0$
$\Rightarrow 7 A^{- 1} = 5 I - A$
$\Rightarrow I = \frac{7}{5} A^{- 1} + \frac{1}{5} A$

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