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**Unformatted text preview: **MAT 442: Test 2 Solutions Throughout, F is a field. 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. 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. (a) Define eigenvalue and eigenvector If T is an operator on a vector space V over F , an eigenvector x ∈ V is a non-zero vector such that T ( x ) = λx for some λ ∈ F . A scalar λ is an eigenvalue if there is a corresponding eigenvector x as above. (b) Define eigenspace If λ ∈ F is an eigenvalue of a linear operator T on a vector space V , the λ- eigenspace is the set { v ∈ V | T ( v ) = λv } . (c) Define characteristic polynomial, including an explanation as to why the definition...

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