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MAT 242 Test 1 SOLUTIONS, FORM A
1. [15 points] Use Cramers Rule to solve the following system of linear equations for x.
3x 2y = 1
2x 2y = 3
Solution:
x=
1
3
3
2
2
2
2
2
=
8
4
=.
10
5
Grading: +5 points for each determinant, +5 points for combining them.
e gff
y d
y w % " xv tA G r A )G u`sI7&f r G75qip&2@0QCGhRQA7)&GfgH&Ge8dG756c29B92761ba7E0`Y7)XVW36UTSR&PI(F&70ED33@797643(0)(&$" 5 1 5 C CA1% B 5Q %2 HG 1 E % C BA212 8 2 51 21% '% # !
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MAT 242 Written Homework #8
1.6, 5.1 5.4
SOLUTIONS
Due: April 17/18
Solve the following problems, showing any necessary work. If you use a graphing calculator, indicate what
calculatons or operations you are having it perform.
11
9
5
4 9 7
1. Let B be t
MAT 242 Test 2 SOLUTIONS, FORM A
1. [30 points] For the matrix A below, nd a basis for the null space of A, a basis for the row space of A,
a basis for the column space of A, the rank of A, and the nullity of A. The reduced row echelon form
of A is the ma
Solutions to Test 3
Problems are identied as (problem number,
problem part, form). Hence, (1aB) is problem 1,
part a, in form B, etc.
(1aA)
75
v1 u
=
= 3,
v1 v1
25
v2 u
50
c2 =
=
= 2,
v2 v2
25
v3 u
75
c3 =
=
= 3,
v3 v3
25
c1 =
11
16
p = c1 v1 + c2 v2 +
MAT 242 Written Homework #7
6.1, 6.2
SOLUTIONS
Due: March 27/28
Solve the following problems, showing any necessary work. If you use a graphing calculator, indicate what
calculatons or operations you are having it perform.
2
1. Let A be the matrix 1
1
00
MAT 242 Written Homework #6
7.3, 4.4
SOLUTIONS
Due: March 20/21
Solve the following problems, showing any necessary work. If you use a graphing calculator, indicate what
calculatons or operations you are having it perform.
1
2
3
1
1
0
1. Let B be the
MAT 242 Written Homework #2
EP1.3, 1.4/H1.2, 2.1
SOLUTIONS
Due: January 30/31
Solve the following problems, showing any necessary work. This includes row operations.
1. Below are three systems of linear equations, in reduced row echelon form.
0
1
0
0
1
0
MAT 242 Written Homework #3
EP1.4, 1.5/H2.2, 2.3
SOLUTIONS
Due: February 6/7
Solve the following problems, showing any necessary work. This includes row operations.
1. [1 point] Find all ordered pairs (x, y ) such that
1
6
x
y
5
y
5
1
=
y
10 x
10 x
6
x
y
MAT 242 Written Homework #4
EP2.2 2.4/H3.1 3.3
SOLUTIONS
Due: February 13/
Solve the following problems, showing any necessary work. This includes row operations.
1. [1 point] Solve the following system of linear equations using Cramers Rule.
4x + y = 1
3
MAT 242 Written Homework #5
4.2, 4.3
SOLUTIONS
Due: March 6/7
Solve the following problems, showing any necessary work. If you use a graphing calculator, indicate what
calculatons or operations you are having it perform.
7
5
1. [1 point] Determine which
MAT 242 Written Homework #1
EP1.2/H1.1
SOLUTIONS
Due: January 23/24
Solve the following problems, showing any necessary work. This includes row operations.
1. Use Gaussian Elimination (with matrices and row operations) and back-substitution to solve the f
EP1.5/H2.3. Inverses of Matrices* There are actually two things we need to do, to make B the X = formula usable. A Find a matrix I such that IM = M for all matrices M (where the product is defined); Given a matrix A, find a matrix C so that CA = I. Then o
EP1.4/H2.1 and H2.2. Matrix Operations
EP1.4/H2.1 and H2.2. Matrix Operations
EP1.4/H2.1 and H2.2. Matrix Arithmetic
Matrices also exist as objects in and of themselves,
and it turns out we can do things to them, which
have algebraic properties. This will
EP1.3/H1.2. Gauss-Jordan Elimination
Remember though that we originally wanted to put
our matrix into the ideal form
100A
0 1 0 B
0 0 1C
Gaussian Elimination takes us to the matrix
1 1 2 3
0
1 2 1
0
0 14
But how do we go further?
The answer is one nal st
MAT 242 A
1
2 1 1. (12) Given that A = 2 0
1 1 -3 4 3 0 4 2 6 4 is row-equivalent to R = 0 1 -3 5 -4 0 0 -1 7 0
0 -2 1 1 0 0 0 0
0 78/7 0 61/7 : 1 -7 0 0
(a) Find a basis for the row space of A.
(b) Find a basis for the column space of
MAT 242 A
1
1 3 6 -1 -5 -2 -4 1 3 0 1. (15) Given that A = 3 6 -2 -4 is row-equivalent to R = 0 4 8 -2 -6 0
2 0 0 0
0 -2 1 -1 : 0 0 0 0
(a) Find all solutions to the homogeneous system of linear equations Ax = 0.
(b) What is the dimen
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MAT 242 Written Homework #2 EP1.3, 1.4/H1.2, 2.1
Due: September 19/20
Solve the following problems, showing any necessary work, including row operations. 1. Below are three systems of linear equations, after performing Gauss-Jordan Elimination on them. (T
MAT 242 Written Homework #3 EP1.4/H2.2
Due: September 26/27
Solve the following problems, showing any necessary work, including row operations. 1. [1 point] Find all ordered pairs (x, y) such that AB = BA, if A= x y 1 5 and B = x 9 y y
EP1.1/H1.0. Introduction to Linear Systems
Algebra deals mainly with equations of one variable.
At the end of MAT 117 (or whatever class you took),
you learned how to solve equations with more than
one variable in them, such as:
x + 3y = 5
4x 2y = 8
How?
EP1.2/H1.1. Matrices and Gaussian Elimination Suppose you need to solve the following system of linear equations: x - y + 2z = 3 2x - 2y + 8z = 22 x - 2y = 2 We're going to be doing elimination over and over, which means writing equations over and over, w
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