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Homework 2
September 4, 2015
Show work to receive credit.
1. (a) Given A defined below, find AT .
(b) What are the dimensions of A and AT ?
(c) How many pivots does A and AT have? (When finding the rref
form of a matrix with more rows than columns, it may be necessary to fill in enough columns with 0 in each element so that the
calculator will perform the operation.)




A=
−3
4
6
0
2
7
3
1
1
3
2
7





2. Defining basic matrix operations, find A−1 . That is, consider [A, I] →
[I, A−1 ]


1 −1 −2


A =  2 −3 −5 
−1
3
5
3. Show that if a square matrix A satisfies A2 − 5A + I = 0, then A−1 =
5I − A.
4. Simplify C −1 A−1 (BAT )T (B −1 )T C
5. Using the inversion algorithm discussed in class to find A−1 . You must
identify E1 , . . . , E5 .


3
5 −6


4 
A =  −6 −9
21 31 −1
6. Find the inverse of each of the following matrices. For each matrix,
how is the stucture of the inverse related to the structure of the original
matrix?


2 −1 0


4 0 ,
A= 3
0
0 5


0 −3 7


4 2 ,
B= 0
3
0 0


6 0 −4


0 ,
C= 0 5
2 0
7
7. Suppose that ~u is a solution to A~x = ~b and that ~v is a solution to
A~x = ~0. Show that ~u + α~v is a solution to A~x = ~b for any value of α.