# HW04 - ï£ ï£¬ ï£¬ ï£¬ ï£¬ ï£­ 1-2 3-12-5 12-14 19-9 22-20...

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EE 203000 Linear Algebra Homework #4 (Due December 4, 2009 BEFORE class) Note: Detailed derivations are required to obtain a full score for each problem. (Total 50%) 1. (14%) Let δ : M 2 × 2 ( F ) F be a function with the following three properties. (i) δ is a linear function of each row of the matrix when the other row is held ±xed. (ii) If the two rows of A M n × n ( F ) are identical, then δ ( A ) = 0. (iii) If I is the 2 × 2 identity matrix, then δ ( I ) = 1. Show that δ ( A ) = det( A ) for all A M 2 × 2 ( F ). ( Hint: First, show that δ ( E ) = det( E ) for any elementary matrix E . Secondly, prove that δ ( EA ) = δ ( E ) δ ( A ) for any elementary matrix E . Finally, by considering the cases where rank( A ) < n and rank( A ) = n , we can show that δ ( A ) = det( A ) for all A M 2 × 2 ( F ) . ) 2. (6%+6%) Compute answers to the following problems. Please show your derivations. (a) Evaluate the determinant of the following matrix:
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Unformatted text preview: ï£« ï£¬ ï£¬ ï£¬ ï£¬ ï£­ 1-2 3-12-5 12-14 19-9 22-20 31-4 9-14 15 ï£¶ ï£· ï£· ï£· ï£· ï£¸ (b) Solve the following systems of linear equations using Cramerâ€™s rule. x 1-x 2 +4 x 3 =-2-6 x 1 +3 x 2 + x 3 = 2 x 1-x 2 + x 3 = 3 . 3. (10%) Let E be an elementary matrix. Show that, for any square matrix B , that det( EB ) = det( E )det( B ) . (Prove directly without using the results of Theorem 4.7.) 4. (10%+4%) Let A * be the Hermitian of the matrix A deÂ±ned by ( A * ) i,j = A j,i for all i , j , where A j,i is the complex conjugate of A j,i (a) Prove that det( A ) = det( A ). (b) A matrix A âˆˆ M n Ã— n ( C ) is called unitary if AA * = I . Prove that, if A is a unitary matrix, then | det( A ) | = 1. 1...
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