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Prove that if A and B are diagonal matrices (of the same size), then AB = BA. Getting Started: To prove that the matrices AB and BA are equal, you need to show that their corresponding entries are equal. (i) Begin your proof by letting A = [aj] and B = [bj] be two diagonal nx n matrices. (ii) The ijth entry of the product AB is Cij k = 1 (iii) Evaluate the entries cj for the two cases i +j and i = j. Cij if i + j Cji = otherwise (iv) Repeat this analysis for the product BA. Find the ijth entry of the product BA. dij = L ajkbki k = 1 O dij = ajkbki k = 1 dij = bikaki k = 1 -Σ |O dij = bjkāki k = 1

Prove that if A and B are diagonal matrices (of the same size), then AB = BA. Getting Started: To prove that the matrices AB and BA are equal, you need to show that their corresponding entries are equal. (i) Begin your proof by letting A = [aj] and B = [bj] be two diagonal nx n matrices. (ii) The ijth entry of the product AB is Cij k = 1 (iii) Evaluate the entries cj for the two cases i +j and i = j. Cij if i + j Cji = otherwise (iv) Repeat this analysis for the product BA. Find the ijth entry of the product BA. dij = L ajkbki k = 1 O dij = ajkbki k = 1 dij = bikaki k = 1 -Σ |O dij = bjkāki k = 1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove that if A and B are diagonal matrices (of the same size), then AB = BA. Getting Started: To prove that the matrices AB and BA are equal, you need to show that their corresponding entries are equal. (i) Begin your proof by letting A = [aj] and B = [bj] be two diagonal nx n matrices. (ii) The ijth entry of the product AB is Cij k = 1 (iii) Evaluate the entries cj for the two cases i +j and i = j. Cij if i + j Cji = otherwise (iv) Repeat this analysis for the product BA. Find the ijth entry of the product BA. dij = L ajkbki k = 1 O dij = ajkbki k = 1 dij = bikaki k = 1 |O dij = bjkāki k = 1 Σ
Transcribed Image Text:Prove that if A and B are diagonal matrices (of the same size), then AB = BA. Getting Started: To prove that the matrices AB and BA are equal, you need to show that their corresponding entries are equal. (i) Begin your proof by letting A = [aj] and B = [bj] be two diagonal nx n matrices. (ii) The ijth entry of the product AB is Cij k = 1 (iii) Evaluate the entries cj for the two cases i +j and i = j. Cij if i + j Cji = otherwise (iv) Repeat this analysis for the product BA. Find the ijth entry of the product BA. dij = L ajkbki k = 1 O dij = ajkbki k = 1 dij = bikaki k = 1 |O dij = bjkāki k = 1 Σ
Evaluate the entries di for the two cases i +j and i= j. dij : if i + j Your answer cannot be understood or graded. More Information otherwise #p Which of the following imply that AB = BA? (Select all that apply.) dij + "p Cii di Cj = 0 Cij = dii Cij = dij
Transcribed Image Text:Evaluate the entries di for the two cases i +j and i= j. dij : if i + j Your answer cannot be understood or graded. More Information otherwise #p Which of the following imply that AB = BA? (Select all that apply.) dij + "p Cii di Cj = 0 Cij = dii Cij = dij
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