Advanced Calculus fi..

8. Prove the following:a) Every square matrix is similar to itself.b) If A is similar to B and B is similar to C, then A is similar to C.Chapter 1 Vectors and Matrices 39*1.12 THE TRANSPOSELet A = (a;,) be an m x n matrix. We denote by A' the n x m matrix B = (bij) suchthat hi, = a,; for i = 1, . . ., n, j = 1, .. . , m. Thus B = A' is obtained from A byinterchanging rows and columns. The following pair is an illustration:The first row of A becomes the first column of A'; the second row of A becomesthe second column of A'. In general, we call A' the transpose of A. We observe thatI' = I . The transpose of a matrix obeys several rules, which we adjoin to our list:24. (A + B)' = A' + B'.25. (cA)' = cA'.26. (A')' = A.27. (A B)' = B'A'.28. If A is nonsingular, then (Ap' )' = (A1)-'To prove Rule 24, we write D = (A + B) = (dij), so thatfor all i and j. Then D' = E = (e,,). where eIj = dj; for all i and j, orpi.; = aJi + b,,.Thus E = A' + B' or D' = A' + B'. The proofs of Rules 25 and 26 are left asexercises (Problem 4 following Section 1.13).To prove Rule 27. we let A be m x p, B be p x n. We then write C = A B =(c,,), D = A' = (d;,), E = B' = (e;,). ThenB'A' = ED = F = (f,,),wheref . -,, , - e,ldl, +... +ejpd,; = bliajl +...+ b PI a JP=~,lb[,+...+a~,b,,~ =c,;(i = 1 ,..., n% j = 1 ,..., m).Hence F = C' or B'A' = (A B)'.