hw3sol - SOLUTIONS ASSIGNMENT 3 2.4.15 Compute the matrix...

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Unformatted text preview: SOLUTIONS: ASSIGNMENT 3 2.4.15 Compute the matrix product 1- 2- 5- 2 5 11 8- 1 1 2 1- 1 . Explain why the result does not contradict Fact 2 . 4 . 9 . 1- 2- 5- 2 5 11 8- 1 1 2 1- 1 = 8- 2- 5- 1- 4 + 5- 16 + 5 + 11 2 + 10- 11 = I 2 . Fact 2 . 4 . 9 only applies if the two matrices being multiplied are n × n . Since these two matrices are 2 × 3 and 3 × 2 they need not be invertible. 2.4.20 True / False: For two invertible n × n matrices A and B , ( A- B )( A + B ) = A 2- B 2 . False. ( A- B )( A + B ) = A 2- B 2 + AB- BA 6 = A 2- B 2 because matrix multiplication is noncommutative. 2.4.23 True / False: ( ABA- 1 ) 3 = AB 3 A- 1 . True. ( ABA- 1 ) 3 = ( ABA- 1 )( ABA- 1 )( ABA- 1 ) = AB 3 A- 1 since matrix multiplication is associative. 2.4.26 Use the given partition to compute the product. Check your work by computing the same product without using a partition. AB = 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 2 2 3 3 4 4 5 0 0 1 2 0 0 3 4 = A 11 A 12 A 22 B 11 B 12 B 22 = A 11 B 11 A 11 B 12 + A 12 B 22 A 22 B 22 1 Using the partition, A 11 B 11 = 1 0 0 1 1 2 3 4 = 1 2 3 4 A 11 B 12 + A 12 B 22 = 1 0 0 1 2 3 4 5 + 1 0 0 1 1 2 3 4 = 3 5 7 9 A 22 B 22 = 1 0 0 1 1 2 3 4 = 1 2 3 4 So AB = 1 2 3 5 3 4 7 9 0 0 1 2 0 0 3 4 . Without using the partition, AB = 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 2 2 3 3 4 4 5 0 0 1 2 0 0 3 4 = 1 2 2 + 1 3 + 2 3 4 4 + 3 5 + 4 0 0 1 2 0 0 3 4 = 1 2 3 5 3 4 7 9 0 0 1 2 0 0 3 4 2.4.41 Consider the matrix D α = cos α- sin α sin α cos α ....
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hw3sol - SOLUTIONS ASSIGNMENT 3 2.4.15 Compute the matrix...

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