Category Archives: linear algebra

Exercise 1.5.8. This exercise continues the discussion in p. 33-34 regarding factorization of a 3×3 matrix A and the proof that A = LU. An alternative proof begins with the fact that row 3 of U is produced by taking … Continue reading →

Exercise 1.5.7. Given the following lower triangular matrices find FGH and HGF. Answer: We have  where so that We also have where so that Thus . UPDATE: Corrected the calculation of HGF; thanks go to Brian D. for pointing out … Continue reading →

Exercise 1.5.6. Given the matrix find , , and . Answer: We have We can square this again to obtain and again to obtain In compting we note that all the entries match those of the identity matrix except for … Continue reading →

Exercise 1.5.5. Given the following system of linear equations find the factors L and U of A and the vector c for which Ux = c. Answer: The 2,1 position of A is already zero, so there’s no need for … Continue reading →

Exercise 1.5.4. Given the matrices use elimination to find the factors L and U for each of the matrices. Answer: For the first matrix we subtract 4 times the first row from the second to obtain U: The other factor … Continue reading →

Exercise 1.5.3. From equations (6) and (3) respectively we have Multiply the two matrices, in both orders. Explain the two answers. Answer: We have and In other words, the product of the matrices is the same in both cases, and … Continue reading →

Exercise 1.5.2. Assume we have a matrix A with What multiple of row 2 of A will elimination subtract from row 3? What will be the pivots? Will a row exchange be required? Answer: We have A = LU where … Continue reading →

Exercise 1.5.1. Given an upper triangular matrix A, under what conditions is A nonsingular? Answer: If A is upper triangular then the diagonal entries of A are the pivots of the corresponding system of linear equations. For A to be … Continue reading →

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 →

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 →