Linear Algebra with Applications (3rd Edition)

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Sol. Use the Gram-Schmidt process: v 1 = 1 1 1 1 ; u 1 = 1 2 1 1 1 1 v 2 = 3 3 - 1 - 1 - ( u 1 · 3 3 - 1 - 1 ) u 1 = 2 2 - 2 - 2 ; u 2 = 1 2 1 1 - 1 - 1 v 3 = 8 0 0 0 - ( u 1 · 8 0 0 0 ) u 1 - ( u 2 · 8 0 0 0 ) u 2 = 4 - 4 0 0 ; u 3 = 1 2 1 - 1 0 0 . 3.Consider the matrixA= 4. Suppose the following information is known about a matrix A A 1 2 1 = 6 1 2 1 , A 1 - 1 1 = 3 1 - 1 1 , A 2 - 1 0 = 3 1 - 1 1 . a) [ 10 marks] Find the eigenvalues of A . Sol. The first two results show that 6 and 3 are eigenvalues of A . The last two results show that there are (at least) two different solutions to the system Ax = 3 - 3 3

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