235HW3 - Math 235: Linear Algebra HW 1 Problem 1.6.1...

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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 row-reduced echelon matrix R which is row-equivalent 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 non-trivial 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 non-zero). 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....
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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.

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235HW3 - Math 235: Linear Algebra HW 1 Problem 1.6.1...

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