# Section 6.3 Exercises

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

1. 

2. 

3. 

4. 

5. 

6. 

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

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

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

1. 

2. 

5. Let A be a nondefective  matrix with diagonalizing matrix X. Show that the matrix  diagonalizes .

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

7. Show that any  matrix of the form



is defective.

8. For each of the following, ...

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