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Unformatted text preview: MAS 4105 Test 3 1. (8 pts) Indicate whether the following are true or false. i. The determinant is defined only for n × n matrices whose entries are real numbers. T F ii. If A is a m × n matrix then rank( A ) < min( m,n ) . T F iii. If n is odd, then det( A ) = det( A T ). T F iv. If ( A  b ) is obtained from ( A  b ) by a finite sequence of elementary column operations, then the systems Ax = b and A x = b are equivalent. T F v. The solution set of any system of m linear equations in n unknowns is a subspace of F n . T F vi. The rank of a matrix is equal to the number of linearly independent rows in the matrix. T F vii. If E is an elementary matrix, then det(E) = ± 1. T F viii. A matrix M ∈ M n × n ( F ) has rank n if and only if det( M ) 6 = 0. T F 2. (10 pts) Let A = 1 2 3 4 5 6 2 8 9 . Calculate the determinant of A using the following two tech niques: i. the definition of the determinant. ii. by placing the matrix in echelon form and using a result for upper triangular matrices. 3. (10 pts) Let V = { ( x 1 ,x 2 ,x 3 ,x 4 ) ∈ R 4 : x 1 + 2 x 3 + 3 x 4 = 0 and x 2 2 x 3 + 4 x 4 = 0 } . i. show that S = ( 5 , 2 , 1 , 1) is an element of V . ii. extend S to form a basis for V . 4. (10 pts) Prove, using the definition of the determinant, that if A is a n × n lower triangular matrix, then det( A ) is the product of its diagonal elements. 5. (10 pts) Given the linear transformation T : R 3 → R 3 defined by...
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 Spring '09
 RUDYAK
 Determinant, Det

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