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Unformatted text preview: 12 3125 1214 199 2220 314 914 15 (b) Solve the following systems of linear equations using Cramer’s rule. x 1x 2 +4 x 3 =26 x 1 +3 x 2 + x 3 = 2 x 1x 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 de±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 ∈ M n × 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
 EE

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