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B) Let D := B + C. Then, (AD)ij = aih dhj = h aih (bhj + chj ) h aih bhj + = h aih chj = (AB)ij + (AC)ij . 15 (Transpose and products) (a) Show that (AB) = B A . (b) Show that (ABC) = C B A . (c) Under what condition is (AB) = A B ? Solution (a) We have (B A )ij = (B )ih (A )hj = h = (B)hi (A)jh h (A)jh (B)hi = (AB)ji . h (b) Let D := BC. Then, using (a), (ABC) = (AD) = D A = (BC) A = C B A . (c) This occurs if and only if AB = BA, that is, if and only if A and B commute. 16 (Partitioned matrix)  1 3 −2 1 0 −1 A= 6 8 0 0 1 4 Let A and B be 3 × 5 matrices, partitioned as    1 −3 −2 2 4 1 2 6 6 , B =  6 2 0 , 1 1 and let C be a 5 × 4 matrix, partitioned as  1 0  0 2  0 C=  −1  3 5 2 −1 5 0 3 0 3 0 1 0 1 2 1 2 0 1 C11 C21 C12 , C22    .

We have 0 ≤ u − λv, u − λv = u, u − λ v, u − λ∗ u, v + |λ|2 v, v . Setting λ := u, v / v, v , the result follows as in the real case. Notice that λ v, u and λ∗ u, v are complex conjugates. 10(c): u+v 2 = u + v, u + v = u, u + u, v + v, u + v, v ≤ u 2 +2 u · v + v 2 = ( u + v )2 , using the Cauchy-Schwarz inequality. Taking the positive square root of both sides yields the result. Notes There are many good introductory texts, see for example Hadley (1961), Bellman (1970), and Bretscher (1997). The reader interested in the origins of matrix theory should consult MacDuffee (1946) or Bretscher (1997).

First, there is Schaum’s Outline Series with four volumes: Matrices by Ayres (1962), Theory and Problems of Matrix Operations by Bronson (1989), 3000 Solved Problems in Linear Algebra by Lipschutz (1989), and Theory and Problems of Linear Algebra by Lipschutz and Lipson (2001). The only other examples of worked exercises in matrix algebra, as far as we are aware, are Proskuryakov (1978), Prasolov (1994), Zhang (1996, 1999), and Harville (2001). Matrix algebra is by now an established field. Most of the results in this volume of exercises have been known for decades or longer.