MATH 1410
Solutions for Homework 5
Submitted Friday, March 1, 2013
(1) Row reduce the following matrices to [reduced row] echelon form. (Make sure you specify your
row operations.)
3
5
6
1
0
5
0
(a) 2 11 11
3 10 20
5
0
Solution:
A matrix is in reduced ro
MATH 1410
Solutions for the Sample Midterm
(1) Find the roots of the following polynomials:
(a) (x 1) x2 + 5x + 6
Solution:
Let a and b be numbers. Then,
(x + a)(x + b) = x2 + ax + bx + ab = x2 + (a + b)x + ab.
This formula can help us factor quadratic (s
MATH 1410
Solutions for Homework 8
Submitted Thursday, March 28, 2013
For all of the questions below, let
A =
and let
1
0
0 2
0
3
0
4
5
0
1
0
= 0 , = 1 , = 0 .
e1
e2
e3
0
0
1
(1) Find [a] vector 1 such that A1 = 1 .
x
x
e
Solution:
Hmmm. . .what size is
Sample Final - Answers
(1) The RREF of the matrix corresponding to this system of equation is
1 0 0 35/11
0 1 0
3/11
0 0 1 29/11
which shows a = 35/11, b = 3/11, and c = 29/11.
(2) A REF of the matrix corresponding to this system of equation is
1 1
k
5
0
MATH 1410
Solutions for Homework 7
Submitted Friday, March 15, 2013
3
1
(1) Let A =
E=
4
2
0
5
, B=
4 2
0
2
, and F = 1
2
1
3
, C =
1
3
5
2
4 , D =
6
0 3
2
1
,
. Compute the indicated matrices (if possible).
(a) A + 2D
Solution:
The addition and scalar mu
MATH 1410
Solutions for Homework 6
Submitted Friday, March 8, 2013
(1)
1
1
(a) Show that R2 = span
1
1
,
.
Solution:
The span of vectors , , . . . (all from Rn ) is the set of all their linear combinations.
v1 v2
vk
n is to say that every vector in Rn i
MATH 1410
Solutions for Homework 1
Submitted Friday, January 18
(1) Solve the following systems of equations:
(a)
5x + y = 59
x + 5y = 31.
Solution:
There are many ways to solve this. Here are four:
Method 1: Substitution (solve for x in the second equati
MATH 1410
Solutions for Homework 4
Submitted Friday, February 15, 2013
(1) Row reduce the following matrices. (Make sure you specify your row operations.)
1
5
6
2
5
(a) 2 11 11
3 10 20
5
Solution:
Let Ri denote row number i. There are three row operation
MATH 1410
Solutions for Homework 3
(The mistake in the fth line on page 10 has been corrected)
Submitted Friday, February 1, 2013
(1) Draw the following vectors in standard position (i.e. starting at the origin).
]
[
= 3
(a) a
1
Solution:
We read the vec
MATH 1410
Solutions for Homework 2
Submitted Friday, January 25
(1) Find all solutions to the following systems of equations:
2x + 3y + 4z = 1
4x 9y + 16z = 1
(a)
3x + 3y
z = 2.
Solution:
Substitution was used more often in the solutions for Homework 1,
Sample, ugly row reduction.
This is a sample of row reduction when numbers dont work out nicely.
Example 1. Row reduce the matrix
9
1
3
5
79 1
3
22
1 4
1
1
3
0
1
3
13
0
0
1
to reduce row echelon form.
Here we go. We start by moving second row to the rst
MATH 1410
Solutions for Homework 9
Submitted Friday, April 5, 2013
(1) Calculate the rank of
2 4
1
2
1 2
0
1
1
1
3 .
4
2
2
4
([The rank] of a matrix is the number of pivot elements in the echelon form of the matrix.)
Solution:
Echelon form? This is a job