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Solution for Let a Be a Square Matrix Such that a 2 − a + I = O , Then Write a − 1 Interms of A. - CBSE (Commerce) Class 12 - Mathematics

Question

Let A be a square matrix such that $A^2 - A + I = O$, then write $A^{- 1}$  interms of A.

Solution

$\text{ Given: } A^2 - A + I = O$
$A^{- 1} \left( A^2 - A + I \right) = A^{- 1} O (\text{ Pre - multiplying both sides because }A^{- 1}\text{ exists})$
$\left( A^{- 1} A^2 \right) - \left( A^{- 1} A \right) + A^{- 1} I = O ( A^{- 1} O = O)$
$\Rightarrow A - I + A^{- 1} = O ( A^{- 1} I = A^{- 1} )$
$\Rightarrow A^{- 1} = I - A$

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Solution Let a Be a Square Matrix Such that a 2 − a + I = O , Then Write a − 1 Interms of A. Concept: Determinant of a Square Matrix.
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