If we define A⁻¹= A^ f(A) where f(A)= |A^T | and A⁻¹ exists then prove that A= adjadj adj....adj A _ ntimes , n is the order of the matrix. Also find the value of n

Step-by-step solution

aligned & Given A⁻¹= A^T f(A) & A^T=f(A) A⁻¹ aligned f(A)= |A^T |=f(A)^n |A⁻¹ | ...(1) But |A⁻¹ |= 1 |A| From (1) 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 ...(2) | adj A adj A|=|A|^ (n-1)^2 ...(3) But from definition |A|= |A^T |=f(A) Hence, from (2) and (3) A= adj A aligned & |A|=f(A)^ n-1 = |A^ |=f(A) & n-1=1 & n=2 aligned