Matrix Algebra I

March 20,2003

Ma 2071

1. if A = compute the product A^t A. What generalization can you make about A^t? (recall that ^t means transpose)

2. a. suppose Q = and R = compute the product QR.

b. What kind of matrices are Q and R? (besides "square" ! )

c. What generalization can you make??

3. a. Suppose D = Please compute D² and D³.

b. What generalization can you make??

4. Suppose A is nxn and x is a column vector with all 0s except the first entry. What can you say about Ax ?

5. Suppose A is nxn and x is a column vector with all 0s except the third entry. What can you say about Ax ?

6. What generalization can you make from problems 4 and 5??

7. Consider the product AB = in two different ways:

a) compute it directly

b) compute 4 times the first column of A plus 5 times the second column of A

c) can you generalize? (you may want to look back at problems 4-6 )

8. a) What is an "upper triangular matrix"?

b) If A is an upper triangular matrix with all nonzero entries on its diagonal, what is the Final Form (RREF) of A ?

9. Suppose A is an nxn matrix and each row of it adds up to the same number, r. (the specific numbers in each row may be different but their total is the same in all cases, r). Suppose u is an nx1 column matrix of all 1's. What can you say about the matrix product Au ? Feel encouraged to make up examples to help decide this.