Download MATH 3260 Exam 3: Definitions, Matrix Multiplication, Determinants, and Inverses and more Exams Linear Algebra in PDF only on Docsity!
MATH 3260 Exam 3 (Version 1) July 7, 2008 S. F. Ellermeyer Name
Instructions. Remember to include all important details of your work. You will not get full credit (or perhaps even any partial credit) if I see gaps in your reasoning. Also, use correct notation and write in complete sentences where appropriate. You may use a calculator on this exam but you may not use any books or notes.
- (DeÖnitions) Use complete sentences to write the following deÖnitions.
(a) Let
A =
a b c d
be a 2 2 matrix. What is meant by the determinant of A? (b) What is the n n identity matrix? (c) What does it mean for an n n matrix, A, to be invertible? (d) What is meant by the inverse of an invertible n n matrix, A? (e) What is an elementary matrix?
(a) Suppose that A is a 3 4 matrix whose columns span V 3 and suppose that C is a 3 3 matrix. Explain how to construct a 4 3 matrix, B, such that AB = C. (This is an essay question. Write carefully and use complete sentences.)
(b) For the matrices
A =
(^5) and C =
use the procedure that you described in part a to Önd a 4 3 matrix, B, such that AB = C.
- Use the algorithm [A I]
I A ^1
to Önd the inverse of the matrix
A =
if it exists. If A ^1 does not exist, then explain how the algorithm tells you this. You may not use determinants or your calculator in doing this problem. You must use the algorithm!
- Compute the determinant 3 1 3 5 1 3 2 3 0 by performing a cofactor expansion along the Örst row. Then compute the determi- nant by performing a cofactor expansion along the Örst column. Show all of your computations.
- Use Cramerís Rule to Önd the solution of the system
3 x 1 2 x 2 = 7 5 x 1 + 6x 2 = 5.
(You must use Cramerís Rule ñnot some other method.)
- Decide whether each of the following statements is true or false.
(a) If A is any n n matrix, then InA = A. (True, False) (b) If A; B; and C are any three n n matrices, then A (B + C) = AB + AC. (True, False) (c) The matrix (^2)
4
is an elementary matrix. (True, False) (d) If A is a 3 3 matrix and the equation
Ax =
has a inÖnitely many solutions, then A must be invertible. (True, False) (e) If A is a square matrix that has a left inverse, then A must also have a right inverse and these left and right inverses must be equal to each other. (True, False) (f) If two rows of a 5 5 matrix, A, are the same, then det (A) = 0. (True, False) (g) If A and B are square matrices of the same size and B is produced by interchanging two rows of A, then det (B) = det (A). (True, False) (h) If A is any square matrix, then det
AT^
= 1= det (A). (True, False) (i) If A is any square matrix, then det (A) det (A ^1 ) = 1. (True, False) (j) Every elementary matrix has determinant 1 or 1. (True, False)
