Section 6.3 Exercises

  1. In each of the following, factor the matrix A into a product XDX1, where D is diagonal:

    1. A=[0110]

    2. A=[5622]

    3. A=[2814]

    4. A=[221012001]

    5. A=[100213111]

    6. A=[121242363]

  2. For each of the matrices in Exercise 1, use the XDX1 factorization to compute A6.

  3. For each of the nonsingular matrices in Exercise 1, use the XDX1 factorization to compute A1.

  4. For each of the following, find a matrix B such that B2=A:

    1. A=[2121]

    2. A=[953043001]

  5. Let A be a nondefective n×n matrix with diagonalizing matrix X. Show that the matrix Y=(X1)T diagonalizes AT.

  6. Let A be a diagonalizable matrix whose eigenvalues are all either 1 or −1. Show that A1=A.

  7. Show that any 3×3 matrix of the form

    [a100a100b]

    is defective.

  8. For each of the following, ...

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