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: If |A| f(A) is the probability that the equation x^ n-1 +(2 n-1) x^n n! +(2 n+3) x^ n-1 +

Question

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: If |A| f(A) is the probability that the equation x^ n-1 +(2 n-1) x^n n! +(2 n+3) x^ n-1 +n=0 has exactly one real root then f(A)^n=x^ n+2 has

Quantrex Academy · Accepted Answer

Correct answer: C. one real root and two imaginary roots

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: If |A| f(A) is the probability that the equation x^ n-1 +(2 n-1) x^n n! +(2 n+3) x^ n-1 +n=0 has exactly one real root then f(A)^n=x^ n+2 has

Options

Correct answer

Step-by-step solution

More from Matrices