ex2 - MAS 3114 Test 2 1. (10 pts) Indicate whether the...

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MAS 3114 Test 2 1. (10 pts) Indicate whether the following are true always, sometimes, or never: i.) If A is a 5 × 7 matrix then its rank is less than 5. always sometimes never ii.) If A and B are n × n matrices such that det A = 0, then AB is singular. always sometimes never iii.) If A and B are n × n matrices then det A + det B = det ( A + B ). always sometimes never iv.) If A is an m × n matrix then the rank of A is equal to Dim Row A T . always sometimes never v.) If A is an n × n triangular matrix then det A 6 = 0. always sometimes never vi.) If A is invertible then det A - 1 = - det A . always sometimes never vii.) If A is n × n and i < j n , then C ij = C ji . always sometimes never viii.) If v 1 and v 2 are vectors in a vector space V , then span { v 1 , v 2 } has dimension two. always sometimes never ix.) If T is a bijective linear transformation from R n to R n and A is the n × n matrix associated with T , then Dim Col A = Dim Nul A . always sometimes never x.) If V is a vector space with dimension p and B is a set of p linearly independent vectors in V , then B is a basis for V . always sometimes never
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2. (5 pts) Prove that the vector 2 1 0 is in the null space of the matrix = - 3 6 - 1 1 - 2 2 2 - 4 5 . 3. (6 pts) Prove that the vector - 2 / 3 - 3 / 2 - 7 / 6 is in the span of the vectors 2 - 3 1 and 5 0 5 .
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4. (10 pts) The matrices A and B given below are row equivalent; find a basis for Col A , Row A , and Nul A . A = - 2 - 5 8 0 - 17 1 3 - 5 1 5 3 11 - 19 7 1 1 7 - 13 5 - 3 B = 1 3 - 5 1 5 0 1 - 2 2 - 7 0 0 0 - 4 20 0 0 0 0 0
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A two ways; first by using cofactor expansion down the first column and and second by using cofactor expansion across the first row (do not use elementary row operations to simplify matrices prior to calculations). A
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ex2 - MAS 3114 Test 2 1. (10 pts) Indicate whether the...

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