quiz3 - det =...

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Quiz #3 1. Label the following statements as being true or false. All matrices in the problems are square. No verification is needed. (a) A matrix M M n × n ( F ) has rank n if and only if det( M ) 6 = 0. (Solution) True : M has rank n if and only if M is invertible. (b) Every system of n linear equations in n unknowns can be solved by Cramer’s rule. (Solution) False : in case the coefficients matrix has determinant 0. (c) If B is a matrix obtained from A by interchanging two rows of A after interchanging two columns of A , then det( B ) = det( A ). (Solution) True : If C is a matrix obtained form A by interchanging two rows(or two columns) of A , then det( C ) = - det( A ). (d) det( - A ) = - det( A ). (Solution) False : det( - A ) = ( - 1) n det( A ) if A is of type n × n . (e) det( A - 1 ) = [det( A )] - 1 . (Solution) True : det( AA - 1 ) = det( A ) det( A - 1 ) = det( I ) = 1. 2. Find the determinants of the following matrices. (a) 1 - 2 4 - 8 1 - 1 1 - 1 1 2 4 8 1 3 9 27 (b) 1 2 3 4 5 6 7 8 0 0 9 10 0 0 11 12 (c) 1 - 1 0 1 0 0 0 0 2 - 1 3 - 1 - 1 0 - 2 1 (d) 1 2 - 4 2 5 0 2 0 8 11 0 0 - 1 2 - 2 0 0 0 - 3 1 0 0 0 0 2 (e) 1 0 1 - 2 3 - 2 1 0 4 - 2 0 5 3 0 1 1 0 1 - 2 0 3 - 1 0 - 6 1 (Solution) (a) Vandermonde determinant.
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Unformatted text preview: det = (-1-(-2))(2-(-2))(3-(-2))(2-(-1))(3-(-1))(3-2) = 240 . (b) det = det 1 2 5 6 det 9 10 11 12 = (-4)(-2) = 8 . (c) The second raw is 0. det = 0. (d) The determinant of an upper triangular matrix is the product of its diagonal entries. det = 12. (e) The fourth column is a multiple of the rst column. det = 0. 1 3. Solve the given system of linear equation using Cramers rule. x 1-x 2 + 4 x 3 =-2-8 x 1 + 3 x 2 + x 3 = 0 2 x 1-x 2 + x 3 = 6 (Solution) Let A = 1-1 4-8 3 1 2-1 1 . Then x 1 = 1 det( A ) det -2-1 4 3 1 6-1 1 =-43 , x 2 = 1 det( A ) det 1-2 4-8 1 2 6 1 =-109 , x 3 = 1 det( A ) det 1-1-2-8 3 2-1 6 =-17 . Therefore ( x 1 ,x 2 ,x 3 ) = (-43 ,-109 ,-17). 2...
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quiz3 - det =...

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