If A is a non-zero square matrix of order n with det (I+A) 0 and A^3=O , where I, O are unit and null matrices of order n n respectively, then (I+A)⁻¹ is equal to

Step-by-step solution

Given, |I+A| O ie, (A+I) is a non-singular matrix. O Null Matrix I Unit Matrix I^3=I A^3=0 A^3+I=O+I A^3+I^3=O+I (A+I) (A^2-A+I )=(O+I)(A+I) (A^2-A+I )=I (O+I=I) Operate (A+I)⁻¹ on both sides (A+I)⁻¹(A+I) (A^2-A+I )=(A+I)⁻¹ I I (A^2-A+I )=(A+I)⁻¹ ( I (A+I)⁻¹=(A+I)⁻¹ ) (A+I)⁻¹= (A^2-A+I ) or (I+A)⁻¹= (I-A+A^2 )