Paragraph: Suppose inverse of a matrix is defined as A⁻¹= A^T f(A) and A⁻¹ exists, where f(A)= |A^T | . Let A be an n^ t^h order matrix then Question: Which of the following statement is correct?

Question

Quantrex Academy · Accepted Answer

Correct answer: A. n =2, ~A = adj ( A )

aligned & A( adj A)=|A| I_n & A^ =f(A) A⁻¹ & |A^T |=f(A)^n |A⁻¹ | f(A)=f(A)^n |A⁻¹ | aligned But |A⁻¹ |= 1 |A| f(A)=f(A)^n 1 |A| |A|=f(A)^ n-1 | adj A|=|A|^ n-1 = [f(A)^ (n-1) ]^ (n-1) =f(A)^ (n-1)^2 ....(1) | adj adj A|=|A|^ (n-1)^2 .....(2) But from defimition |A|= |A^ |=f(A) aligned & |A|=f(A)^ = |A^ |=f(A) & n-1=1 & n=2 aligned From (1) and (2) A= adj A aligned & f(x)=x^3+ 5 x^2 2 +6 x+2 & f^ (x)=3 x^2+5 x+6 aligned aligned & D 0 & f^ (x) 0 x R aligned Graph for f(x) f ( x ) has only one real root aligned & p=1

Paragraph: Suppose inverse of a matrix is defined as A⁻¹= A^T f(A) and A⁻¹ exists, where f(A)= |A^T | . Let A be an n^ t^h order matrix then Question: Which of the following statement is correct?

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