SHOW YOUR WORK!!

A/ Use Row Reduction to solve if possible:

1)

3)

(16)

5) For which values of k does this system have nontrivial solutions?

(4)

6) For which values of a will this system have:

a) no solutions b) a unique solution c) infinite solutions?

(6)

B/

1) find AB 2) find BA 3) find (2BA + 3C) 4) write B t .

5) Find the inverse for the only one of these 3 matrices that has an inverse.

(10)

.

6) Prove that if AB and BA are both defined, then both AB and BA
are square matrices.(A and B are not the matrices above)

(4)

C/

1) Find an elementary matrix E 1 which adds twice row 2 to row 3 of A.

2) Find the inverse of E 1.

3) Find A –1 using elementary matrices.

4) Using your answer from #3, solve :

(10)

5) Determine conditions on a, b, and c so that this system has a solution.

(4)

D/

1) Given that find the determinants of:

(6)

.

D/

2) Use the determinant and the adjoint of A to find A –1

(5)

3) Use Cramer's Rule to solve

(3)

4) For which values of k does A fail to be invertible if

(2)

E/

1) Given that u = , and v = , find:

a) 2u – 3v b) || u || c) cos A if A is the angle between u and v d) projv u

(6)

2) Find an equation for the plane that contains point P (1, –1, 2)
and the line x = t, y = 1 + t, z = –3 + 2t.

(3)

3) Show the line x = –t + 5, y = 2t – 3, z = –5t – 1 is parallel to the plane –3x + y + z = 9.

(2)

4) Find an equation for the plane that passes through (2, 4, –1)
and contains the line of intersection of the planes x – y – 4z = 2
and –2x + y + 2z = 3.

(5)

F/

1)

a) Find a basis for the nullspace of A.

b) Find a basis for the row space of A.

c) Find a basis for the column space of A.

d) What is the nullity of A?

e) What is the rank of A?

f) Does belong to the column space of A? Why or why not?

g) Describe geometrically the nullspace of A. (What kind of subspace of R3 is it?)

(14)

TOTAL (100)

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