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The solutions to exam 3 of math 3260, which covers various topics including matrix definitions, matrix multiplication, determinants, and linear transformations. Students are expected to understand concepts such as determinants of 2x2 matrices, identity matrices, invertible matrices, and their inverses, as well as the effects of multiplying matrices on the left by elementary matrices.
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MATH 3260 Exam 3 (Version 2) Solutions 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.
(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
(^5) and C =
use the procedure that you described in part a to Önd a 4 3 matrix, B, such that AB = C.
Solution: For the unknown matrix,
B = [b 1 b 2 b 3 ] ,
it must be true that
AB = [Ab 1 Ab 2 Ab 3 ] = [c 1 c 2 c 3 ] = C.
Thus, to Önd B, we must solve all of the matrix equations Ab 1 = c 1 , Ab 2 = c 2 , and Ab 3 = c 3. (Solutions to all of these equations exist because the columns of A span V 3 .) Since all of these matrix equations have the same coe¢ cient matrix
A = [a 1 a 2 a 3 a 4 ] ,
we can Önd B in ìone fell swoopîby rowñreducing the ìtripleñaugmentedîmatrix
[a 1 a 2 a 3 a 4 c 1 c 2 c 3 ].
Example: For the matrices, A and C, given in part b above we have
and we see from this that one possible B such that AB = C is given by
This B was obtained by assigning x 4 = 0 in each equation Abi = ci. There are inÖnitely many other B such that AB = C. For example, note that
also works.
to Önd the inverse of the matrix
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! Solution:
Thus
A ^1 =
7 2
3 2
1 2