MATH 2050 Linear Algebra (Section 4)

Department of Mathematics and Statistics
2009 Winter

Term Test 2

2009 March 13
[Determinants, Eigenvalues, Eigenvectors, Diagonalization]

  1. Find the value of the determinant of each of the following matrices.
    1.   A = [ 3 0 0 0 ] [ x -2 0 0 ] [ 17 y 1 0 ] [ pi 42 -6 4 ]

      [3]

    2.   B = (a 5x5 matrix with rows 2 & 4 the same)

      [3]

    3.   C = [ 4 -2 1 5 ] [ 2 9 0 3 ] [ 0 2 0 0 ] [ 5 8 0 7 ]

      [5]

  1. Find all eigenvalues and corresponding eigenvectors of the matrix
             
    Hence write down both the matrix   P   that diagonalizes   A   and the diagonal matrix   D = P^-1 AP.
    [Note:   you do not have to evaluate the matrix product   P^-1 AP.]

    [13]

  1. Evaluate   det([2A]^(-1) [3A]T), where   A   is any invertible (2×2) matrix.

    [6]

      BONUS QUESTION:

  1. Find   (3,4) element of T-inverse   (the entry in row 3, column 4 of the inverse matrix   (3,4) T-inverse)
    for the matrix   T = [ 1 2 3 4 ] [ 0 5 6 7 ] [ 0 0 2 8 ] [ 0 0 2 8 ] [ 0 0 0 1 ].

    [+5]

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    Created 2009 03 05 and most recently modified 2009 03 05 by Dr. G.H. George