assign8

# assign8 - a A = 1 0 0 2 3 0 1 4 1 b B = 2-2 1-2 3 1 4-1 2 6...

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Math 136 Assignment 8 Due: Wednesday, Mar 24th 1. For each of the following matrices, ﬁnd the inverse, or show that the matrix is not invertible. a) A = 1 - 1 2 3 1 5 2 2 3 . b) B = 1 - 1 0 2 0 1 1 0 2 - 2 3 5 1 0 1 3 . 2. Let B = 2 - 1 1 0 1 1 1 - 1 - 1 . Find B - 1 and use it to solve B~x = ~ d , where ~ d = (4 , - 2 , 3). 3. a) Prove that if A and B are n × n matrices such that AB is invertible, then A and B are invertible. b) Give an example of 2 × 3 matrix A , and 3 × 2 matrix B such that AB is invertible. Are A and B invertible? 4. Write the 4 × 4 elementary matrices that correspond to each of the following elementary row operations. a) add -2 times the third row to the second. b) interchange the ﬁrst row and the fourth row. c) multiply the second row by 3. 5. Write each of the following matrices and their inverses as a product of elementary matrices.
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Unformatted text preview: a) A = 1 0 0 2 3 0 1 4 1 b) B = 2-2 1-2 3 1 4-1 2 6. Find an LU-factorization of A = 2-2 1-2 3 1 4-1 2 , and use it to solve A~x = 2-4 6 using a forward substitution followed by a back substitution. 7. Find an LU-factorization of A = 2-4 4-2 6-9 7-3-1-4 8 . 8. Let A be an invertible n × n matrix and let { ~v 1 , . . . ,~v n } be a basis for R n . Prove that { A~v 1 , . . . , A~v n } is also a basis for R n . 1...
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