Review Problem Set 1 (Systems of Linear Equations and Matrices)
Math 2270-001 (Summer 2005)
Do Not Submit
1
DISCLAIMER:
The types of questions that may appear in tests and the nal exam
are not limited to those in the review problem sets!
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ID:
Code Name:
Instructions
1. Print your name, ID number, and code name above. Your code name will be used for posting grades on the course
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2. If you supplied a code name in the last Test, please leave the code name entry blank.
3. Time allowed: 1 hour.
4. Solve all problems and show all work for full credit.
5. No calculators are allowed.
6. The total number of points in this test is ??. It will be graded out of ??.
1. (10 pts) Compute the following matrix product:
"
2
#6
1 2 3 4 66
6
5 6 7 8 64
8
6
4
2
7
5
3
1
3
7
7
7
7
7
5
Review Problem Set 1 (Systems of Linear Equations and Matrices)
Math 2270-001 (Summer 2005)
Do Not Submit
2. (10 pts) Find the row-reduced echelon form of the following matrix:
2
1
4 2
1
1 1
2 1
1 1
3
1
35
3
2
Review Problem Set 1 (Systems of Linear Equations and Matrices)
Math 2270-001 (Summer 2005)
Do Not Submit
3. (10 pts) Solve the following system of equations for , , and :
x
x
+2 =1
z
;
y
z
3 + =2
y
z
;
x
+ =3
y
:
3
Review Problem Set 1 (Systems of Linear Equations and Matrices)
Math 2270-001 (Summer 2005)
Do Not Submit
4. (10 pts) Find the inverse of the following matrix:
2
6
6
6
6
6
4
0
0
0
1
0
1
0
0
1
0
0
0
0
0
1
0
3
7
7
7
7
7
5
4
Review Problem Set 1 (Systems of Linear Equations and Matrices)
Math 2270-001 (Summer 2005)
Do Not Submit
5. (10 pts) Prove that if a matrix has an inverse, then the inverse is unique.
5
Review Problem Set 1 (Systems of Linear Equations and Matrices)
Math 2270-001 (Summer 2005)
Do Not Submit
6. (10 pts) Find the determinant of the following matrix:
2
1
6 0
6
4 0
1
1
1
0
0
0
0
1
1
3
1
0 77
15
0
6
Review Problem Set 1 (Systems of Linear Equations and Matrices)
Math 2270-001 (Summer 2005)
Do Not Submit
7. (10 pts) Compute the determinant of the following matrix:
2
6
6
6
6
4
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
2
3
4
5
6
3
7
7
7
7
5
7
Review Problem Set 1 (Systems of Linear Equations and Matrices)
Math 2270-001 (Summer 2005)
Do Not Submit
8. (10 pts)
a) Give an example of two nonzero matrices whose sum is not dened.
b) Give an example of two nonzero matrices whose product is not dened.
8
Review Problem Set 1 (Systems of Linear Equations and Matrices)
Math 2270-001 (Summer 2005)
Do Not Submit
9
9. (10 pts) Let be the matrix such that left multiplication by on 3 matrices corresponds to the following
row operation: Add to the second row two times the rst row. What is ?
M
M
n
M
Review Problem Set 1 (Systems of Linear Equations and Matrices)
Math 2270-001 (Summer 2005)
Do Not Submit
10. (10 pts) Prove that matrix multiplication is associative using summation notation.
10
Review Problem Set 1 (Systems of Linear Equations and Matrices)
Math 2270-001 (Summer 2005)
Do Not Submit
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Review Problem Set 1 (Systems of Linear Equations and Matrices)
Math 2270-001 (Summer 2005)
Do Not Submit
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