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Monthly Archives: August 2010
Linear Algebra and Its Applications, exercise 1.4.24
Exercise 1.4.24. The following matrices can be multiplied using block multiplication, with the indicated submatrices within the matrices multiplied together: Provide example matrices matching the templates above and use block multiplication to multiply them together. Provide two example templates for … Continue reading
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Linear Algebra and Its Applications, exercise 1.4.23
Exercise 1.4.23. Find the results of squaring, cubing, and in general raising to the power of n the following matrices: Answer: For the matrix A we have By induction we have for all n. For the matrix B we have … Continue reading
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Linear Algebra and Its Applications, exercise 1.4.22
Exercise 1.4.22. The xy plane can be rotated through an angle by the following matrix: Show that . Hint: use the identities for and . Compute . Answer: (a) We have We can simplify the final matrix using the following … Continue reading
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Linear Algebra and Its Applications, exercise 1.4.21
Exercise 1.4.21. An alternative way to compute the matrix product AB is as the sum where is the ith column of A, is the ith row of B, and the product is a matrix. Provide an example showing the procedure … Continue reading
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Linear Algebra and Its Applications, exercise 1.4.20
Exercise 1.4.20. If A and B are two nxn matrices with all entries equal to 1, what are the entries of their product AB? Use the following summation formula to find the answer: Also, if C is a third nxn … Continue reading
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Linear Algebra and Its Applications, exercise 1.4.19
Exercise 1.4.19. Given matrices A and B, which of the following matrices are equal to ? Answer: First, since matrix addition is commutative we have as well as So the matrices referenced in (a) and (d) above are equal to … Continue reading
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Linear Algebra and Its Applications, exercise 1.4.18
Exercise 1.4.18. Given arbitrary nxn matrices A and B, show that the first column of AB is the same as A times the first column of B. Hint: Let x = (1, 0, …, 0) be a column vector with … Continue reading
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Linear Algebra and Its Applications, exercise 1.4.17
Exercise 1.4.17. Assume A is a 2×2 matrix and further assume that AB = BA for any 2×2 matrix B, including the matrices Show that a = d and that b and c are zero. Answer: We have and Since … Continue reading
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Linear Algebra and Its Applications, exercise 1.4.16
Exercise 1.4.16. For the following matrices verify that (EF)G = E(FG). Answer: We have and We also have and NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its … Continue reading
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Linear Algebra and Its Applications, exercise 1.4.15
Exercise 1.4.15. Given the matrix E given by and an arbitrary 2×2 matrix A, describe the rows of the product matrix EA and the columns of AE. Answer: For EA we have The first row of EA is a linear … Continue reading
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