MA552. Midterm exam.This takehome exam due October 17, 20061. Let A denote an n × n matrix with elements belonging to a field K. Let P denote thecharacteristic polynomial of A and assume P (0) 6= 0.Show that A has an inverse A−1and that the characteristic polynomial R of A−1is definedbyR(λ) =(−1)nλnP (0)P1λHint: one may, for example, examine the determinant of A(A−1− λI). If K is a subfield ofC, one may use a triangular matrix B that is similar to A.2. Consider the matrix A:(a) A =1 3 03 −2 −10 −1 1(b) A =1 1 0−1 2 11 0 1(1) Find the characteristic values and the eigenvectors of A(2) Find a nonsingular matrix T such that the matrix T−1AT will b e a diagonal matrix D.Find the matrix T−1and check your calculations by taking the product T−1AT , writedown D.3. Let A be 2×2 matrix w ith complex elements and with a double characteristic value λ. Showthat there exists a matrix B similar to A (that is, B = T−1AT ) and equal to one of twomatrices λ 00 λ!, λ 10 λ!Find Bn14. Given the matrix A:A =8 −1 −5−2 3 14 −1 −1(1) Find the characteristic values and the eigenvectors of A.(2) Find a basis s uch that the transformed matrix of A, which we denote by B, will betriangular and find the matrix B.(3) Show that among all bases that satisfy the condition of (2), there exists at least onesuch that the matrix B has one and only one nonzero element off the principal diagonaland that this term is equal to 1. Find the corresponding matrix B0.(4) Find Bn0. Show without carrying the calculations, how one can find