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# Su11c - MAS 4105 Test 3(12 pts 1 In the following let T be...

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MAS 4105 Test 3 (12 pts) 1. In the following, let T be a linear operator on a vector space V and let β be a basis for V . Also let A, B, C, D, E M n × n . a. If matrix A is invertible and matrix B is not, what is the maximum possible rank of AB ? b. Let U be a linear operator on R 3 defined by U a b c = a + b + c b + c c then the vector - 3 0 0 is an eigenvector for U . What is the eigenvalue associated with the eigenvector? c. The function f ( t ) = det([ T ] β - tI) is called the of T . d. If C = - 1 2 1 0 - 3 2 4 - 2 - 2 , write the terms resulting from the calculation of the determinant by performing a cofactor expansion down the third column (do not evaluate!). e. The determinant of matrix C is equal to -2. If matrix C is obtained from matrix D by interchanging two rows of D and if matrix D is obtained from matrix E by multiplying one row of matrix E by -4, what is the determinant of matrix E ? f. The system Ax = b is consistent if rank( A | b ) is equal to ?

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(8 pts) 2. Prove that the system of linear equations Ax = b has a solution if and only if b R ( L A ) .
(8 pts) 3. Given the matrix A = 1 2 1 1 3 2 1 2 2 , solve the equation Ax = - 1 2 - 3 by first finding A - 1 and then using this matrix to obtain the solution.

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(8 pts) 4. A matrix M M n × n ( R ) is called skew-symmetric if M t = - M . Prove that if M is skew symmetric and n is odd, then M is not invertible. (Note you are free to use the result from an assigned homework problem in your proof.)
(8 pts) 5. Let

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Su11c - MAS 4105 Test 3(12 pts 1 In the following let T be...

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