# Suppose that A and B are square matrices of the same order. Show by example that (A + B)…

Math 112 Homework Assignment 2 due date: January 27, 2011 1. [2pts each] (a) Suppose that A and B are balance matrices of the corresponding direct. Show by illustration that (A + B)2 = A2 + 2AB + B2 want not stop. Can you reinstate the over individuality after a while a amend individuality. (b) Suppose that A, B are 2 × 2 matrices after a while AB = 0. Does it thrive that A or B is the naught matrix? Give a demonstration or a counterexample. (c) Suppose that A, B and C are non-naught 2 × 2 matrices after a while AB = AC. Does it thrive that B = C? Give a demonstration or a counterexample. 2. [2pts each] (a) Let A be an m×n matrix. Let AT be its reverse. Then AT is an n×m matrix. Consider the emanation AAT . What is the direct of AAT ? Suppose that AAT = 0. What could be the matrix A satisfying this quality? (b) Let A =24 -3 13 -135. What is the direct of the matrix (rA)T (rA) for any r 2 R? Find all values r 2 R such that (rA)T (rA) = 1. (c) If A is a balance matrix, is A - AT besides symmetric? (d) If A and B are symmetric matrices of the corresponding direct, are AB and BA besides symmetric? 3.(a) [1pt] Let A = · 3 -5 -1 2 ¸, B =24 1 -1 2 -2 3 -335and C = ·2 0 -4 5 -2 6 ¸. Check that (AC)B = A(CB). (b) [2pts] Suppose that A and B are matrices, and the emanation BAB is well-defined. What can you say encircling the directs of A and B? 1 4. [1pt each] Let A =24 3 1 -4 6 9 -2 -1 2 135and B =24 2 -4 6 3 -5 8 -2 0 1035. Verify that (a) tr(A + B) = tr(A) + tr(B). (b) tr(AT ) = tr(A). (c) tr(AB) = tr(BA). Here trstands for the investigate of a balance matrix. 2

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