310f04_1 - AX = Y has no solution. 3. [12 points] Consider...

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Problem 1 2 3 4 5 Bonus: Total: Points 15 13 12 5 5 10 50+10 Scores Mat 310 – Linear Algebra – Fall 2004 Name: Id. #: Lecture #: Test 1 (September 24 / 50 minutes) There are 5 problems worth 50 points total and a bonus problem worth up to 10 points. Show all work. Always indicate carefully what you are doing in each step; otherwise it may not be possible to give you appropriate partial credit. 1. [15 points] Consider the homogeneous system of linear equations x 1 + x 2 + 2 x 3 - 2 x 4 = 0 x 1 - 5 x 2 - x 3 + 7 x 4 = 0 x 1 - x 2 + x 3 + x 4 = 0 (a) [3 points] Write down the matrices A , X , and O for which the system is in matrix form AX = O . (b) [6 points] Using the Gauss-Jordan algorithm, compute the row-reduced echelon matrix R which is row equivalent to A . (c) [6 points] Use (b) to find all solutions of the above system.
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2. [14 points] Let A = 2 - 2 1 3 - 1 2 1 - 3 0 , X = x 1 x 2 x 3 , Y = y 1 y 2 y 3 . (a) [10 points] Apply row reduction to the augmented matrix of A and determine exactly for which triples ( y 1 ,y 2 ,y 3 ) the system AX = Y has a solution. (b) [3 points] Write down an explicit right hand side column Y for which the system
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Unformatted text preview: AX = Y has no solution. 3. [12 points] Consider A = 2 3 3 4 . (a) [8 points] By any method, argue that A is invertible and compute A-1 . (b) [4 points] Write A as a product of elementary matrices. 4. [5 points] Find an example of 2 2 matrices A,B for which it is not true that ( A + B ) 2 = A 2 + 2 AB + B 2 . [Can you give a condition for A and B , so the last matrix formula would hold? Youll get 5 extra points for the right answer.] 5. [5 points] Let A be an n n matrix. Show that A is invertible if and only if A 2 is invertible. Bonus Problem [up to 10 points] Consider the equation X 2 =-I for 2 2 matrices. There are solutions with real coecient. Discover as many as you can. Can you nd innitely many? Give some more additional solutions with complex coecients. [You might also want to observe that if P is invertible 2 2, then if X solves, so does PXP-1 . Why?]...
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310f04_1 - AX = Y has no solution. 3. [12 points] Consider...

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