MAT444test2-soln

MAT444test2-soln - MAT 442: Test 1 Solutions 1. Read these...

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MAT 442: Test 1 Solutions 1. Read these instructions carefully. For each type of row operation give the following: (a) A description of what it does (so we know which row operation you are talking about) (b) The determinant of a corresponding elementary matrix E (c) State whether or not E t is an elementary matrix (d) State whether or not E is invertible, and how we know that. Note, you do not have to know the number which goes which each type of row operation, but you have to cover all types. Type I operations: a) Swap row i with row j with i 6 = j . b) det( E ) = - 1 c) Yes it is an elementary matrix, in fact, E t = E d) Yes it is invertible because we can undo the operation by repeating it: E 2 = I Type II operations: a) Multiply row i by k F where k 6 = 0 b) det( E ) = k c) Yes it is an elementary matrix, in fact, E t = E d) Yes, because we can undo the operation by multiplying the same rwo by 1 /k , which exists since k 6 = 0 Type III operations: a) Add k times row i into row j where i 6 = j (i.e., R j R j + kR i ) b) det( E ) = 1 c) Yes it is an elementary matrix – the transpose corresponds to adding k times row j into row i d) Yes it is invertible because we can undo the operation by adding - k times row i into row j , which corresponds to multiplying by another elementary matrix 2. Suppose T ∈ L ( V,W ) and U ∈ L ( W,Z ) are linear transformations where V , W , and Z are finite-dimensional vector spaces. (a)
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MAT444test2-soln - MAT 442: Test 1 Solutions 1. Read these...

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