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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- Spring '11
- Linear Algebra, 10%, 6%, 4%, Î´