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 133 Midterm Solutions. Problem 1: The problem is asking to find all the values of a and b so that a given system system of equations has either no, a unique, or infinitely many solutions. Perform Gaussian rowreduction: 2 1 1 1 2 a 4 3 1 6 11 b 1 2 a 2 1 1 4 3 1 11 6 b 1 2 a 5 2 a + 1 5 4 a 1 11 16 b 44 1 2 a 5 2 a + 1 2 a 2 11 16 b 28 . In the first step we interchange the first two rows; in the second step we subtract twice the first row from the second row, and subtract four times the first row from the third row; in the third step we subtract the second row from the third row. In principle we can also divide the second row by 5 to get an explicit 1 in the second column. If 2 a 2 6 = 0 (in other words a 6 = 1) then we can divide the third row by 2 a 2 and obtain a 1 in the third column. Hence the system has a unique solution regardless of what values are on the right. (Indeed, we can uniquely solve for x 3 using the third equation, then we can uniquely solve for x 2 using the second equations, and finally we can uniquely solve for x 1 using the first equations.) If 2 a 2 = 0 AND b 6 = 28 then the system is inconsistent: the third equation is of the form 0...
View Full
Document
 Spring '08
 KLEMES
 Math

Click to edit the document details