MA 2321 Exam 1 Solutions, Spring 2010
1. (10 points) Let A be a 5 5 matrix. If A has 3 pivot columns, answer the following questions.
(a) Rank A is 3.
(b) The dimension of Nul A is equal to 2.
(c) The matrix A (can/cannot) cannot be reduced to the 5 5 ide
Answers to True/False questions Section 5.1 Problem 21
1. True, By denition of eigenvalue.
2. True, A is not invertible if 0 is one of the eigenvalues and if zero is one of the eigenvalues, A
is not invertible (if and only if means the statement is true b
Answers to True/False questions Section 3.1 Problem 39
1. True, we nd the cofactors which are nothing but determinants of matrices one order size
smaller than what you start with.
2. True, obvious by denition (and the many examples we have seen)
Answers t
Answers to True/False questions Section 2.1 Problem 15
1. False, Matrix multiplication involves row times column and not column times column.
2. False, The other way around. See the blue box on page 110.
3. True, Since these products are dened already (se
Answers to True/False questions Section 1.1 Problem 23
1. True, Read the relevant line on page 7 just below that box. This means you can do a row
operation and change it back if necessary.
2. False, It has 5 rows.
3. True, substituting the solution for th
Answers to True/False questions Section 6.1 Problem 19
1. True, By denition of dot product of a vector with itself and length of vector.
2. True, See properties of dot product.
3. True, see the picture (Fig 5) on page 379. This is the condition from which
MA 2321 Quiz 1 Spring 2010, Solutions
1. (10 points) Answer the following questions.
(a) An 8 3 matrix has 8 rows and 3 columns.
(b) The maximum number of pivot positions an 8 3 matrix can have is 3.
(c) If a 3 7 matrix has 2 pivot columns, it has 2 basic
MA 2321 Sample Exam Solutions(For Test 1)
1. This exam is prepared so that you can and should complete doing all problems in less than 50
minutes.
2. You must show work to all problems (except the objective type).
3. If a problem has multiple parts, answe
MA 2321 Quiz 4 Solutions Spring 2010
1. Find a basis for the eigenspace for the following matrices (for the specic eigenvalue given) (You
must show work supporting your answer)
(a)
50
21
, = 1
Here = 1 is an eigenvalue. You DO NOT have to verify it again.
MA 2321 Quiz 2 Solutions Spring 2010
1. (10 points) Answer the following questions.
(a) The product
2 3
80
is
4
7
13
32
(b) Every homogeneous system is consistent. This statement is True/False
True, The right hand side of a homogeneous equation
is always