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MHT CET202526 Apr 2025Evening ShiftMathematicsMatricesActual

If A is a matrix of order 2 and I is the identity matrix of order 2 such that A ^2-4 ~A +3 I =0 then ( A +3 I )⁻¹=

Options

  1. AA 24 - 7 24 I
  2. BA 21 - 7 21 I
  3. C7 I 24 - 1 24 ~A
  4. DA-3 I

Correct answer

C. 7 I 24 - 1 24 ~A

Step-by-step solution

Given the matrix equation A^2 - 4A + 3I = 0 , we seek the inverse of A + 3I . Let X = A + 3I , so that A = X - 3I . Substituting into the given equation: (X - 3I)^2 - 4(X - 3I) + 3I = 0 Expanding, since X and I commute: X^2 - 6X + 9I - 4X + 12I + 3I = 0 Combining terms: X^2 - 10X + 24I = 0 Rewriting to isolate 24I : 24I = 10X - X^2 Factoring the right side: 24I = X(10I - X) Multiplying both sides on the right by X⁻¹ : 24X⁻¹ = 10I - X Solving for X⁻¹ : X⁻¹ = 1 24 (10I - X) Substituting X = A + 3I : (A + 3I)⁻¹ = 1 24

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