1. Consider the equations 2x + y = 2 3x + 2y = 2 . a) Write this system of equations in the form A x = b. b) Recall the \"row picture\" of the equations: each equation cuts out a line in R2 . Is the intersection of the lines a point, a line, or empty? How many solutions does the system of equations have? c) What is the \"pivot\" in the first step of elimination? (Remember: the pivot is the coefficient of the first variable, in the first equation). What is the multiplier ` ? (Remember: the first step of elimination subtracts ` times the first equation from the second equation.) d) Write down the triangular system that results from elimination. Then solve it for x, y. 2. If elimination succeeds for a given system A x = b, how many solutions does it have? If elimination \"fails,\" how many solutions can it have? How is success or failure of elimination related to the independence of the columns of A? What about independence of the rows of A? 2. Consider the equations x + z = 0 x + y = 0 y + z = 0 a) Write this system of equations in the form A x = b. b) Recall the \"row picture\" of the equations: each equation cuts out a plane in R3 . Is the intersection of the three planes a point, line, plane, or empty? How many solutions does the system of equations have? c) Perform step 1 of elimination: remove the first variable (x) from the second and third equations. d) Perform step 2 of elimination: remove y from the third equation. e) Why does elimination \"fail\"? f) What is the full solution (the full set of solutions) to the system? 3. Repeat steps (a)-(f) of problem 2 for x + z = 0 x + y = 0 y + z = 1. a 1 2 4. Consider a system A x = b with A = a a 3. For which three values of a will a a a elimination \"fail\"? 1 5. Consider 2x + y + z = 1 x + y z = 0 x + 3y + z = 2 . a) Write down the augmented matrix (A b) that represents this system. b) Set = 1 and use elimination to solve the system. c) For what value of does elimination require a reordering of the rows? What is the solution in this case? d) For what value of does elimination \"fail\"? How many solutions does the system have in this case: zero or infinity? 6. For the following pairs of matrices, compute AB and BA. (Notice that in special cases AB = BA, completely different.) \u0012 but \u0013 in general \u0012 AB \u0013 and BA are \u0012 \u0013 \u0012 \u0013 x 0 a 0 0 1 a b a) A = , B= b) A = , B= 0 y 0 b 1 0 \u0012 \u0013 \u0012 \u0013 \u0012 \u0013 \u0012 c d\u0013 1 0 1 0 1 x 1 0 c) A = , B= d) A = , B= a 1 0 1 a 1 x 1 1 0 0 1 0 0 1 0 0 1 0 0 e) A = x 1 0 , B = a 1 0, f) A = x 1 0 , B = a 1 0 . y 0 1 b 0 1 y 0 1 0 b 1 7. Compute AB for \u0012 \u0013 1 0 2 1 1 0 a) A = 0 1 , B = 1 2 0 1 23 x \u0001 c) A = y , B = a b . z 8. Consider 1 0 2 A = 3 1 5 , 0 4 1 a \u0001 b) A = x y z , B = b c 2 1 0 B = 1 2 1 . 0 0 1 Compute the product AB in two different ways: a) by multiplying rows of A and columns of B (to obtain each entry of AB, one by one); or b) by multiplying columns of A and rows of B. For part (b), that if we think of A = (u v w) as containing three column remember x vectors and B = y as containing three row vectors, we should have AB = ux + vy + wz. z 2 9. Recall that raising a matrix to an integer power is n times z }| { AA A |n| times An = z }| { 1 1 1 A A A I defined as if n > 0 if n < 0 if n = 0 . \u0012 \u0013 1 1 Let A = . Find A2 , A3 , A4 , A5 . Do you see a pattern? What is An for an arbitrary 1 0 positive integer n? (Hint: Fibonacci numbers!) 10. Consider x + y + = 1 2y z + 3w = 0 2x + z = 5 x 3y + w = 2 a) Write down the augmented matrix (A b) that represents the system. b) Which elimination matrix E1 implements the first step of elimination (removing x from equations 2,3,4) by multiplying (A b) on the left? c) Which elimination matrix E2 implements the second step (removing y from equations 3, 4) by subsequently multiplying on the left? d) After steps 1 and 2, a reordering of rows is required. Which permutation matrix P implements this? e) Write down the triangular form of the augmented matrix, and finish solving the system. 3