Test 2. Fall 1996.

The instructions are standard. Show your work and justify each step in the "proof" problem.

1 (20 points). Prove that if A and B are square matrices of the same size, B is not 0, and AB=0, then det(A)=0.

2 (15 points). Let L be the set of all pairs (x,y) where x, y are real numbers. Define two operations + and scalar multiplication, on L:

(x,y)+(x',y')=(x+y', x'+y)
k(x,y)=(kx,ky)

Check if L is a vector space.

3 (15 points). Find the determinant of the following n by n matrix:

                        2 3 0 0 ..... 0
                        0 2 3 0 ..... 0
                        ...............
                        3 0 0 0 ......2

4 (20 points). Find x from the following system of equations using Cramer's method:

                        x+y+z=5
                       2x+3y-z=1
                        x-y+z=3

(you should use Cramer's method, any other method is not allowed.)

5 (20 points). Find the matrix of the linear operator in R² which first rotates through 45 degrees counterclockwise and then reflects about the x-axis.

6 (10 points). Find the standard matrix of the following linear transformation:

(x,y,z) ---> (5x+y-z, x-z, y+z)