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test1-sp94

# test1-sp94 - P then M is row equivalent to P 4(8 points...

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MAS 4105 Test 1 January 21, 1994 Show all your work on the paper provided. 50 points total. Your work should be written in a proper and coherent fashion. 1. (8 points) Solve the system of equations: x + 2 y + z + 3 w = 0 x - y + w = 0 y - z + 2 w = 0 2. (9 points) Consider the matrix B = - 1 0 2 - 1 . (a) Calculate B - 1 . (b) Calculate | 3 B T | . (c) Write B as a product of elementary matrices. 3. (8 points) (a) Define what it means to say that a matrix B is row equivalent to a matrix A . (b) Use the definition of row equivalence to prove that if M is row equivalent to N , and N is row equivalent to
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Unformatted text preview: P , then M is row equivalent to P . 4. (8 points) Prove the following. If X and Y are n × n matrices and X Y = I then X is invertible and X-1 = Y . 5. (9 points) Let A be an n × n matrix. (a) Deﬁne what it means to say that A is nonsingular . (b) Prove the theorem that states that if A is nonsingular, then its inverse is unique. 6. (8 points) A matrix W is called skew-symmetric if W T =-W . Prove that if B is a square matrix, then B-B T is skew-symmetric. 1...
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