Name: MATH 3195 :: Fall 2011 :: Exam 2. No document, no calculator, 1h00. Explanations and justifications are expected for full credit.

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1 Name: MATH 3195 :: Fall 2011 :: Exam 2 No document, no calculator, 1h00. Explanations and justifications are expected for full credit. 1. ( 4 pts) Say which matrix is in row echelon form and which is not. If the matrix is in row echelon form then give the associated reduced row echelon form and enumerate the free and leading variables. are nonzero entries. (a) (b) , 2. (6 pts) For each problem, give the solution set in parametric form. (a) { x1 + 3x x 4 + 6x 5 = 2, x 3 + 2x 4 + x 5 = 1. (b) { x1 + 3x 2 + 4x 3 + 2x 4 + 3x 5 = 0, x 2 + x 3 + 2x 4 + 2x 5 = (6 pts) For each matrix equality, give the elementary coefficient matrix that enables to go from the right to the left. You can either multiply by the elementary matrix on the left or on the right but not on both sides. (a) ( L 2 L 2 3L 1 ) and ( L 3 L 3 6L 1 ) =

2 (b) ( C 2 C 2 5C 1 ) and ( C 3 C 4 ) = (10 pts) Give the set of solution for the two 2x2 linear systems of equations below using three different ways. 1) Using row echelon form. (Reduced or not.) 2) Using the formula for the inverse of a 2 by 2 matrix, find the inverse of A. Then use the inverse of A to solve the system Ax = b. 3) Using Cramer s Rule. Please be clear. If a technique does not work, say it. { 5x + 6y = 12 (a) 3x + 4y = 6 { x + 2y = 2 (b) 2x + 4y = 4 5. (10 pts) Give the set of solutions. (a) { x + 2y = 1 x + ky = 1 { 2x + y = 1 (b) 8x + 4y = k 6. (5 pts) Find the equation of the line which goes by the points (1,2) and (3,-1). 7. (8 pts) Find the cartesian equation for the circle determined by the three points: (-1,5), (5,-3) and (6,4). What are the center and the radius of this circle? 8. (8 pts) Compute the determinant of (6 pts) Compute the inverse of (6 pts) With your own words, explain why the process you have used works. (This is a discussion about elementary matrices.) 10. (6 pts) We consider the linear system Ax = b. (Matrix A is square, n-by-n.) We consider three different cases, in each case, we want to know whether it has (a) a unique solution, (b) an infinite number of solutions or (c) no solution. Justification required. Case 1: b = 0 and det(a) = 0. Case 2: b is anything, A is row equivallent to the identity matrix. Case 3: b is anything, A is not invertible (so A 1 does not exist)..

3 11. (6 pts) Consider Ax = b where A is m-by-n, what is the size of x? what is the size of b? 12. (4 pts) Consider Ax = b where A is square (n-by-n) and invertible, how would you find x using the backslash notation? 13. (8 pts) Prove that the inverse of a diagonal matrix is a diagonal matrix.

4 1. ( 4 pts) Say which matrix is in row echelon form and which is not. If the matrix is in row echelon form then give the associated reduced row echelon form and enumerate the free and leading variables. are nonzero entries. (a) (b) ,

5 2. (6 pts) For each problem, give the solution set in parametric form. (a) { x1 + 3x x 4 + 6x 5 = 2, x 3 + 2x 4 + x 5 = 1. (b) { x1 + 3x 2 + 4x 3 + 2x 4 + 3x 5 = 0, x 2 + x 3 + 2x 4 + 2x 5 = 1.

6 3. (6 pts) For each matrix equality, give the elementary coefficient matrix that enables to go from the right to the left. You can either multiply by the elementary matrix on the left or on the right but not on both sides. (a) ( L 2 L 2 3L 1 ) and ( L 3 L 3 6L 1 ) = (b) ( C 2 C 2 5C 1 ) and ( C 3 C 4 ) =

7 4. (10 pts) Give the set of solution for the two 2x2 linear systems of equations below using three different ways. 1) Using row echelon form. (Reduced or not.) 2) Using the formula for the inverse of a 2 by 2 matrix, find the inverse of A. Then use the inverse of A to solve the system Ax = b. 3) Using Cramer s Rule. Please be clear. If a technique does not work, say it. { 5x + 6y = 12 (a) 3x + 4y = 6 (b) { x + 2y = 2 2x + 4y = 4

8 5. (10 pts) Give the set of solutions. (a) { x + 2y = 1 x + ky = 1 (b) { 2x + y = 1 8x + 4y = k

9 6. (5 pts) Find the equation of the line which goes by the points (1,2) and (3,-1).

10 7. (8 pts) Find the cartesian equation for the circle determined by the three points: (-1,5), (5,-3) and (6,4). What are the center and the radius of this circle?

11 8. (8 pts) Compute the determinant of

12 9. (6 pts) Compute the inverse of (6 pts) With your own words, explain why the process you have used works. (This is a discussion about elementary matrices.).

13 10. (6 pts) We consider the linear system Ax = b. (Matrix A is square, n-by-n.) We consider three different cases, in each case, we want to know whether it has (a) a unique solution, (b) an infinite number of solutions or (c) no solution. Justification required. Case 1: b = 0 and det(a) = 0. Case 2: b is anything, A is row equivallent to the identity matrix. Case 3: b is anything, A is not invertible (so A 1 does not exist).

14 11. (6 pts) Consider Ax = b where A is m-by-n, what is the size of x? what is the size of b? 12. (4 pts) Consider Ax = b where A is square (n-by-n) and invertible, how would you find x using the backslash notation?

15 13. (8 pts) Prove that the inverse of a diagonal matrix is a diagonal matrix.

16 Name: MATH 3195 :: Fall 2011 :: Exam a 9b