exam2solutions

# exam2solutions - Math 217, Linear Algebra, Fall 2002 Exam 2...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 217, Linear Algebra, Fall 2002 Exam 2 Solutions 1.(8pts) Let A be the matrix A = 1 2 3 4 5 6 7 8 9 10 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 60 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 . It turns out that A x = is true for both x = 1-2 1 and x = 65-80 5 4 3 2 1 . Does A x = b have a solution for all possible b ∈ R 9 ? Explain. (Hint: don’t use your calculator). We know that rank( A ) + dim(Nul( A )) = 10, the number of columns. Each of the given vectors is an element in Nul( A ) = { x ∈ R 10 | A x = } , and they are clearly independent (because there are only two, we may check to see that they are not multiples of each other, which is true), so dim(Nul( A )) ≥ 2. This means that rank( A ) ≤ 8, and in particular that A can have at most 8 pivots. This means that in row reduced echelon form, A has a row of zeros, and thus the equation A x = b will not be consistent for all b ∈ R 9 . 2.(10pts) Suppose that the matrix B = 1 2 0 0 0 3 0 0 0 0 3 2 0 0 0 1 was obtained from A by the following row and column operations (performed in the order listed). 1. I replaced row 4 with row 4 plus 2 times row 3. 2. I replaced col 1 with col 1 minus 3 times col 4. 3. I swapped rows 2 and 4. 4. I multiplied row 3 row by the scalar 3. 5. I replaced row 4 with row 4 plus 4 times row 3. 6. Finally I swapped columns 3 and 1. What is det A ? Show your work....
View Full Document

## 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.

### Page1 / 5

exam2solutions - Math 217, Linear Algebra, Fall 2002 Exam 2...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online