math339midreview - False The dimension of the column space...

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Math 339 Midterm Review (1) Given the matrix A = 1 0 0 0 1 1 0 1 1 . (a) Find the eigenvalues and eigenvectors. (b) Diagonalize A , that is, find matrices P and D such that A = PDP - 1 . (c) Compute A 7 using the decomposition in part b. (2) (a) Find the LU decomposition A = 4 3 - 5 - 4 - 5 7 8 6 - 8 (b) Use the LU factorization to solve Ax = b where b = [2 - 4 6] T . (3) Find the characteristic polynomial of A = - 1 0 1 - 3 4 1 0 0 2 . (4) Let T be a linear transformation defined by T ( x 1 , x 2 , x 3 ) = ( x 1 - 5 x 2 + 4 x 3 , x 2 - 6 x 3 ) . Determine if T is one-to-one and onto . (5) (a) True or False . The dimension of the column space of
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Unformatted text preview: False . The dimension of the column space of A is rank A . (b) True or False . The dimension of Col A is the number of pivot columns. (c) True or False . If A T is not invertible, then A is not invertible. (d) True or False . If AB is invertible, then so is B . (6) Let A and P be square matrices, with P invertible. Show that det ( PAP-1 )= det ( A ). (7) Find the inverse of the matrix A = 1 2 1 3 4-3 8 . 1...
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