This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 235: Linear Algebra HW 1 Problem 1.6.1 Problem 1.6.1 Let A = 1 2 1 0 1 3 5 1 2 3 5 Find a rowreduced echelon matrix R which is rowequivalent to A and an invertible 3x3 matrix P such that R = PA . To do this row reduce as follows: 1 2 1 0 1 0 0 1 3 5 0 1 0 1 2 1 1 0 0 1 → 1 2 1 0 1 0 0 2 4 5 1 1 0 4 0 1 1 0 1 → 1 0 3 5 1 0 0 2 4 5 1 1 0 0 8 11 1 2 1 → 1 0 3 5 1 0 1 2 5 2 1 2 1 2 0 0 1 11 8 1 8 1 4 1 8 → 1 0 0 7 8 3 8 1 4 3 8 0 1 0 1 4 1 4 1 4 0 0 1 11 8 1 8 1 4 1 8 This gives us that R = 1 0 0 7 8 0 1 0 1 4 0 0 1 11 8 and that P = 1 8 3 2 3 2 2 1 2 1 Problem 1.6.7 Let A be an nxn matrix. Prove the following two statements: (a) If A is invertible and AB = 0 for some nxn matrix B then B = 0 AB = 0 ⇒ A 1 AB = A 1 ⇒ IB = B = 0 (a) (b) If A is not invertible, then there exists some nxn matrix B such that AB = 0 but B 6 = 0 Since A is not invertible we know that Ax = 0 has a nontrivial solution (where both 0 and x are nx1 column vectors). Let B be the nxn matrix where every column of B is the column vector x . Matrix multiplication will show that the product of these matrices is the matrix where very column is the 1xn column vectors consisting of all zeros. In other words AB = 0, where in this case 0 is the nxn 0 matrix. (b) Problem 1.6.8 Let A = a b c d . Prove, using elementary row operations, that A is invertible if and only if ( ad bc ) 6 = 0 First note that we can assume that a and c are not both zero since if they were then ad bc would equal zero, and the left column of A would be all zero which would clearly make A not invertible. We therefore assume a 6 = 0 (note if a was zero then we simply exchange our 2 rows since c would then be nonzero). Row reducing we get the following: A = a b c d ⇒ 1 b a c d , this operation is defined since a 6 = 0, → 1 b a d bc a = 1 b a ad bc a Now we can proceed by row reducing to 1 b a 1 → 1 0 0 1 if and only if ad bc 6 = 0 Problem 1.6.8 continued on next page...Problem 1....
View
Full
Document
This note was uploaded on 11/14/2010 for the course PHYCS 498 taught by Professor Aa during the Spring '10 term at University of Illinois, Urbana Champaign.
 Spring '10
 aa

Click to edit the document details