Multivariable Calculus and Matrix Algebra, MATH2010

Third Exam, Thursday, April 4, 2002.

You may use one sheet of handwritten notes, but no other sources. The exam consists of four questions, and lasts fifty minutes. Please work all four problems. Please show all work clearly and in reasonable detail. Answers without appropriate supporting work or requested explanations may not receive full credit.

1.

(25 points) Consider the three vectors

$$u = \left[ \begin{array}{r} 1 \\ 0 \\ 2 \\ -1 \end{array} \... ...left[ \begin{array}{r} -1 \\ a \\ 3 \\ 2 \end{array} \right].$$

For what value(s) of a are these vectors linearly independent?

2.

Let A be a 3 x 3 matrix and let

$$u = \left[ \begin{array}{r} -7 \\ 4 \\ 1 \end{array} \right... ...b = \left[ \begin{array}{r} 0 \\ 2 \\ -1 \end{array} \right].$$

(a)

(15 points)  Use the facts that Au=e₁, Av=e₂, and Aw=e₃ to find A^-1.

(b)

(10 points) Use the result of part 2a to solve the system of equations Ax=b.

3.

Let

$$A = \left[ \begin{array}{rr} 3 & 1 \\ 5 & 2 \end{array} \ri... ...\begin{array}{rr} 2 & 0 \\ -4 & 3 \end{array} \right], \;\;\;$$

(a)

(10 points) Find A^-1.

(b)

(15 points) Find a matrix C satisfying A^-1CA^T = BA^T.

4.

Let A be the matrix

$$A := \left[ \begin{array}{rrrrrr} 1 & 2 & 1 & 3 & -2 & 0 \\ -1 & -2 & -1 & -1 & 4 & 1 \end{array} \right] .$$

(a)

(15 points) Find a basis for the null space of A.

(b)

(10 points) What is the rank of A?