Multivariable Calculus and Matrix Algebra, MATH 2010

You may use four sheets of handwritten notes, but no other sources. Answer the first three problems and any three of the remaining six problems. Please show all work clearly and in reasonable detail. Answers without appropriate supporting work or requested explanations may not receive full credit. No books or calculators allowed.

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Answer completely questions 1, 2, and 3.

1.
(15 points) Consider the system of equations Ax=b, where

\begin{displaymath} A = \left[ \begin{array}{cccc} 1 & 0 & a & 3 \\ 0 & 1 & 2 ... ... g = \left[ \begin{array}{r} p \\ q \\ r \end{array} \right]. \end{displaymath}

(a)
For what value(s) of a and c does Ax=b have infinitely many solutions?
(b)
For what value(s) of a and c is the system Ax=b inconsistent?
(c)
Let a=c=1. For what value(s) of p, q, and r does the system Ax=g have a unique solution?

2.
(20 points) Let

\begin{displaymath} A = \left[ \begin{array}{rrr} 2 & 0 & 1 \\ 0 & 1 & 0 \\ 1 ... ...= \left[ \begin{array}{r} 2 \\ 2 \\ -2 \end{array} \right]. \end{displaymath}

(a)
(5 points) What are the eigenvalues of A?
(b)
(10 points) Find a matrix S that diagonalizes A.
(c)
(5 points) Calculate A7 b. What is $\lim_{k \rightarrow \infty} A^k b$?

3.
(20 points)
(a)
(10 points) Let

\begin{displaymath} u = \left[ \begin{array}{r} 1 \\ 3 \\ -2 \end{array} \right... ... w = \left[ \begin{array}{r} a \\ b \\ c \end{array} \right]. \end{displaymath}

For what value(s) of a, b, and c are the three vectors u, v, and w orthogonal?

(b)
(10 points) Let

\begin{displaymath} p = \left[ \begin{array}{r} 1 \\ 3 \\ -2 \\ 2 \end{array} \... ...left[ \begin{array}{r} 7 \\ -1 \\ 2 \\ -9 \end{array} \right] \end{displaymath}

be a basis for a subspace W of $I \!\! R^4$. Find an orthogonal basis for W.

 

Answer any three of the following problems 4 through 9
4.
(15 points)
(a)
The temperature at the point (x,y) on a metal plate is

\begin{displaymath} T = \frac{2x}{x^2+y^2}. \end{displaymath}

i.
(5 points) Find the direction of greatest increase in temperature from the point (4,3).
ii.
(5 points) Find the directions of no change in temperature from the point (4,3).

(b)
(5 points) Find the absolute extrema of the function f(x,y)=(2x+y-1)2 over the triangular region in the xy-plane with vertices (0,0), (1,0), and (0,1).

5.
(15 points) The center of mass of the square lamina with vertices at (0,0), (2,0), (0,2), and (2,2) with constant density is at the point (1,1).
(a)
(9 points)  Make a conjecture about how the center of mass $(\bar{x},\bar{y})$ will change for the nonconstant densities $\rho(x,y)$ given below. Mark your estimate on the picture.
i.
$\rho(x,y)=3y$.


\begin{picture} (200,120) \put(10,10){\vector(1,0){100}} \put(10,10){\vector(0,1... ...0} \put(87,0){2} \put(0,87){2} \put(105,0){$x$ } \put(0,105){$y$ } \end{picture}

ii.
$\rho(x,y)=x^2+2y^2$.


\begin{picture} (200,120) \put(10,10){\vector(1,0){100}} \put(10,10){\vector(0,1... ...0} \put(87,0){2} \put(0,87){2} \put(105,0){$x$ } \put(0,105){$y$ } \end{picture}

iii.
$\rho(x,y)=(2-x)y$.


\begin{picture} (200,120) \put(10,10){\vector(1,0){100}} \put(10,10){\vector(0,1... ...0} \put(87,0){2} \put(0,87){2} \put(105,0){$x$ } \put(0,105){$y$ } \end{picture}

(b)
(6 points) Find the mass and center of mass for one of the densities in part 5a.
6.
(15 points) Use Green's Theorem to find the area inside the ellipsoid x2+9y2=9.

7.
(15 points) Let

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

(a)
(5 points) Find a basis for the range (column space) of A.

(b)
(5 points) Find a basis for the nullspace of A.

(c)
(5 points) What are the nullity and rank of A?
8.
(15 points) Answer the following either TRUE or FALSE. Give short justifications for your answers. Each part is worth 3 points.
(a)
$\textstyle \parbox{6in}{Let $u$\space and $v$\space be two linearly independent... ...n I \!\! R^3$\space such that $\{u,v,w\}$\space is a basis for~$I \!\! R^3$ .}$         
(b)
  $\textstyle \parbox{6in}{Let $A=\left[\begin{array}{rrr}1 & 2 & 3 \\ -1 & 2 & 0 \\ 1 & 3 & 0 \end{array} \right]$ . The determinant of $A$\space is 15.}$         
(c)
% latex2html id marker 600 $\textstyle \parbox{6in}{If $A$\space is as in part~\... ...c{1}{5} \\ \frac{1}{3} & \frac{1}{15} & -\frac{4}{15} \end{array} \right]$ .}$         
(d)
$\textstyle \parbox{6in}{Let $A$\space and $B$\space be $n \times n$\space matri... ...f $A$\space and $B$\space are both nonsingular then $AB$\space is nonsingular.}$         
(e)
$\textstyle \parbox{6in}{The vector $u=\left[ \begin{array}{r} 1 \\ i\end{array}... ...nvector of $A=\left[ \begin{array}{rr} 2 & -1 \\ 1 & 2 \end{array} \right]$ .}$         

9.
(15 points) Let

\begin{displaymath} A = \left[ \begin{array}{cc}6&4-2a\\ 4-2a&3a \end{array}\rig... ..., \qquad v = t \left[ \begin{array}{r}1\\ -2\end{array}\right] \end{displaymath}

for parameters a, s, and t.
(a)
(5 points) Show that u and v are eigenvectors of A for any choice of a.
(b)
(5 points) What is the largest eigenvalue of A when a=3? What is the largest eigenvalue of A when a=1?

(c)
(5 points) What value of a minimizes the largest eigenvalue of A?


 
John E. Mitchell
2003-11-17