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1. Let A = [a)ax5, B = [bj]ax5, C = [cy]sx2, D = [dy]ax2, and E = [eg]sx4- Determine which of the following matrix expressions are defined. For those which are defined, give the size of the resulting matrix. (a) BA (b) AC + D (c) AE + B (d) AB + B (e) E(A+ B)

1. Let A = [a)ax5, B = [bj]ax5, C = [cy]sx2, D = [dy]ax2, and E = [eg]sx4- Determine which of the following matrix expressions are defined. For those which are defined, give the size of the resulting matrix. (a) BA (b) AC + D (c) AE + B (d) AB + B (e) E(A+ B)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 27EQ
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UNIVE20649237564332Y pdf 2 DEPARTMENT OF MATHEMATICS AND STATISTICS MM201 Linear Algebra and Differential Equations 1 Exercises: Chapter 1 1. Let A = [a]ax5, B = [byjlax5, C = [cy]5x2, D = [dij]4x2, and E = [eij]5x4- Determine which of the following matrix expressions are defined. For those which are defined, give the size of the resulting matrix. (b) AC + D (c) AE + B (h) (A" + E)D. (а) ВА (d) AB + B (e) E(A+ B) (f) E(AC) (g) ETA 2. Consider the matrices 3 0 -1 2 A = B = C = 6 1 3 -1 1 2 152 D= -1 0 1 3 2 4 E = 413 Compute the following (where possible). (a) 4E – 2D (b) 2B – C (c) AB (d) BA (e) (3E)D (f) (AB)C (g) A(BC) (h) (DA)" (i) ATD" (j) tr(4ET – D) (k) tr(CC"). 3. Solve the following matrix equation for a, b, c and d: a - b b+e 3d +e 2a – 4d [::)- 4. Let B = C = Verify that (AB)-1 = B-' A- and (ABC)- = C- B-'A-!. MM201 Linear Algebra and Differential Equations 5. Let D be a non-singular matrix such that -3 (7D)-- Find D. 6. Evaluate tr(0,xn) and tr(1), and prove that (a) tr(A+ B) = tr(A) + tr(B), (b) tr(AB) = tr(BA), (c) tr(AA) = tr(A) for any n x n matrices A, B and A ER. 7. In each of the following, determine whether the matrix is in echelon form, reduced echelon form, both or neither. 1EU 3 0 10 05 0 01 3 0!10 1031 (a) (b) (c) 0 0 0 0 1
Transcribed Image Text:UNIVE20649237564332Y pdf 2 DEPARTMENT OF MATHEMATICS AND STATISTICS MM201 Linear Algebra and Differential Equations 1 Exercises: Chapter 1 1. Let A = [a]ax5, B = [byjlax5, C = [cy]5x2, D = [dij]4x2, and E = [eij]5x4- Determine which of the following matrix expressions are defined. For those which are defined, give the size of the resulting matrix. (b) AC + D (c) AE + B (h) (A" + E)D. (а) ВА (d) AB + B (e) E(A+ B) (f) E(AC) (g) ETA 2. Consider the matrices 3 0 -1 2 A = B = C = 6 1 3 -1 1 2 152 D= -1 0 1 3 2 4 E = 413 Compute the following (where possible). (a) 4E – 2D (b) 2B – C (c) AB (d) BA (e) (3E)D (f) (AB)C (g) A(BC) (h) (DA)" (i) ATD" (j) tr(4ET – D) (k) tr(CC"). 3. Solve the following matrix equation for a, b, c and d: a - b b+e 3d +e 2a – 4d [::)- 4. Let B = C = Verify that (AB)-1 = B-' A- and (ABC)- = C- B-'A-!. MM201 Linear Algebra and Differential Equations 5. Let D be a non-singular matrix such that -3 (7D)-- Find D. 6. Evaluate tr(0,xn) and tr(1), and prove that (a) tr(A+ B) = tr(A) + tr(B), (b) tr(AB) = tr(BA), (c) tr(AA) = tr(A) for any n x n matrices A, B and A ER. 7. In each of the following, determine whether the matrix is in echelon form, reduced echelon form, both or neither. 1EU 3 0 10 05 0 01 3 0!10 1031 (a) (b) (c) 0 0 0 0 1
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