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NATIONAL UNIVERSITY OF SINGAPORE SEMESTER 2, 2016/2017 MA1101R Linear Algebra Homework Assignment 2 Please write your name, student card number and tutorial group number on the answer script, and submit during the lecture on 16th February 2017 (Thursday). 1. For each of the following matrices, (i) Use Gauss-Jordan elimination to find its inverse. (ii) Write down the elementary row operation used in each step, and the determinant of the corresponding elementary matrix. (iii) Find the determinant of the given matrix. (a) A = (aij )3×3 , where aij = 1 ; i+j (b) B = (bij )3×3 , where bij = |i + j| − |i − j|. 1 1 1 1 0 1 1 1 . Find adj(A) by using Definition 2.5.24. 2. Let A = 0 0 1 1 0 0 0 1 3. Use Cramer’s rule to solve the following linear system: 3x − 2y + 5z = 1 4x − y + z = 2 −2x + y + 2z = 3. 4. For each pair of S1 and S2 , determine whether (i) span(S1 ) ⊆ span(S2 ), and (ii) span(S2 ) ⊆ span(S1 ). (a) S1 = {(1, 0, 1, 0), (1, 2, 1, 2), (3, 1, 2, 1)}, S2 = {(1, 1, 1, 1), (2, 1, 2, 1), (3, 1, 3, 1)}. (b) S1 = {(3, −2, 5, 1), (4, −1, 1, 2), (−2, 1, 2, 3)}, S2 = {(9, −4, 4, 0), (0, −2, 11, 3), (−1, −2, −1, 9)}. 5. Let A be a square matrix. Prove that there exists a nonzero square matrix B such that AB = 0 if and only if det(A) = 0. [Hint: Use Theorems 2.4.7 and 2.5.19. Note that matrices can be multiplied in blocks; see equation (2.2).] 6. Let A be a square matrix of order n that has exactly one nonzero entry in each row and each column. Let D be the diagonal matrix whose ith diagonal entry is the nonzero entry in the ith row of A. 1 MA1101R LINEAR ALGEBRA HOMEWORK ASSIGNMENT 2 0 0 a1 a2 [For example, if A = 0 0 0 0 0 0 a4 0 2 a1 0 0 0 0 a2 0 0 0 , then D = 0 0 a 0 .] a3 3 0 0 0 a4 0 0 Prove that (i) A = DP for a permutation matrix P . (ii) A−1 = AT D −2 . [Hint: Refer to Homework Assignment 1, Problem 6. For (i), show that a permutation matrix P can be obtained from A by a series of elementary row operations. For (ii), use the result that P P T = I which was proved in Homework Assignment 1.]