exam1solutions - Math 217, Linear Algebra, Fall 2002 Exam...

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Unformatted text preview: Math 217, Linear Algebra, Fall 2002 Exam 1, October 4, 2002 Name: Solutions 1.(5pts) Let A be a 6 5 matrix. What must a and b be in order to define T : R a R b by T ( x ) = A x ? If we are trying to compute A x then x must be a length 5 vector. The result of A x is a length 6 vector. So a = 5 and b = 6. 2.(5pts) Give an example of a 2 2 matrix A which has the following three properties: 1) A 6 = , 2) A 6 = I 2 , and 3) A T = A . Try 1 0 0 0 on for size. In fact, any matrix a b b c will work as long as if b = 0 then both a and d are not both 1 or both zero. 3.(10pts) Note that the matrix A = 4 5 6 7 8 7 6 5 6 7 8 9 is similar to the matrix B = 4 5 6 7- 3- 6- 9 6 7 8 9 . Write down an elementary matrix E such that EA = B . What is E- 1 ? You can find an elementary matrix E such that EA = B by performing the same row operation on the identity as you did on A . That is, replace Row(2) of the identity with Row(2)- 2Row(1). Doing this yields the matrix E = 1 0 0 0- 2 1 0 0 0 1 0 0 0 1 . The matrix E- 1 is such that E- 1 E = I 4 . So we find E- 1 by performing the same row operation on the identity as we would perform on E to obtain I 4 . That is, Row(2) Row(2)+2Row(1) transforms E to I 4 , thus Row(2) Row(2)+2Row(1) transforms I 4 to E- 1 = 1 0 0 0 2 1 0 0 0 0 1 0 0 0 0 1 . 4.(5pts) Consider the matrix A = 1 2 3 20 2 3 4 21 . . . . . . 19 20 21 39 ....
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This note was uploaded on 04/13/2010 for the course ECON 330 taught by Professor Minetti during the Fall '08 term at Michigan State University.

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exam1solutions - Math 217, Linear Algebra, Fall 2002 Exam...

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