Category Archives: linear algebra

Exercise 1.4.22. The x-y 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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

Exercise 1.4.14. Show example 2×2 matrices having the following properties: A matrix A with real entries such that A nonzero matrix B such that Two matrices C and D with nonzero product such that CD = -DC Two matrices E … Continue reading →

Exercise 1.4.13. Provide an example of multiplying two 3×3 triangular matrices, to confirm the general case that the product of triangular matrices is itself a triangular matrix. Prove the general case based on the definition of matrix multiplication. Answer: An … Continue reading →