If A and B are square matrices of order 3 such that 2(A+B)=A^T+B^T+3 I and A A^T=4 I , then find the value of det. (12 A⁻¹-B A^T+I ) . [Note: I is an identity matrix of order 3 and P^T denotes the transpose matrix of matrix P .]

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Correct answer: 125

aligned & A+B=3 I A A^T+B A^T=3 A^T A A^T=3 A^T-B A^T 4 I=3 A^T-B A^T & 12 A⁻¹-B A^T+I=5 I & det (12 A⁻¹-B A^T+I )=125 aligned

If A and B are square matrices of order 3 such that 2(A+B)=A^T+B^T+3 I and A A^T=4 I , then find the value of det. (12 A⁻¹-B A^T+I ) . [Note: I is an identity matrix of order 3 and P^T denotes the transpose matrix of matrix P .]

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