HELP Matrices problem

that's question is asking given the A is n x n ( x = multiply ) square that satisfies A^2 - 3A - 2I = O , where I is n x n identity matrix and O is zero matrix. find ( A - I) ^-1

Let : B = A - I so that A = B + I.

∵ A² - 3A - 2I = O

∴ (B+I)² - 3(B+I) - 2I = O

∴ (B+I)(B+I) - 3B - 3I - 2I = O

∴ ( B² + BI + IB + I² ) - 3B - 5I = O

∴ B² + B + B + I - 3B - 5I = O

∴ B² - B - 4I = O

∴ B² - B = 4I

∴ B² - BI = 4I

∴ B ( B - I ) = 4I

∴ B [ (1/4)( B - I ) ] = I

∴ Bֿ¹ = (1/4)( B - I )

∴ ( A - I )ֿ¹ = (1/4)[ ( A - I ) - I ]

∴ ( A - I )ֿ¹ = (1/4)( A - 2I ) ....................... Ans.
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