Week 2 Homework
Abdul-M
ps07_f16.pdf
Problem Set 7 - Matrix Inverses
Learning Objectives:
• You should know the definition of an invertible linear transformation/matrix and understand what the inverse linear transformation/matrix represents.
• You should be able to determine whether a given matrix is invertible and, if so, compute its inverse.
• You should understand the relationship between whether vectors ~v1, . . . ,~vm form a basis of Rm and whether the matrix
[ ~v1 · · · ~vm
] is invertible.
Note about the reading: As you already know, reading mathematics requires a careful eye; in this section especially, most of the theorems are stated only for square matrices. It’s worth thinking about this more: what happens in the analogous situations if the matrices are not square?
1. (a) Determine whether each of the following matrices is invertible; if so, find its inverse by hand, and check that you’ve done so correctly by multiplying your answer by the original matrix. (Does the order of multiplication matter?)
Note: For 2 × 2 matrices, we highly recommend that you memorize Theorem 2.4.9.
i. [
5 −3 1 7
] ii. [ −2 6
3 −9
] iii.
1 1 −1−5 −4 4
2 2 −1
(b) Solve the linear system
1 1 −1−5 −4 4
2 2 −1
~x =
10
7
. (Rather than using Gauss-Jordan, can you use
your answer to (a)iii?)
2. Bretscher #2.4.104
3. Suppose we have an unknown linear transformation T : R3 → R2, and we know that T
1−1
0
= [1
0
] ,
T
2−1
4
= [2
0
] , and T
35
3
= [4
0
] . The vectors
1−1
0
, 2−1
4
, and
35
3
form a basis of R3, so
we know that the given information determines T completely; that is, we should be able to find the matrix of T from this information. You’ve done this before in problems like Problem Set 4, #3, but now that we know about matrix products and inverses, we can find the matrix of T more efficiently.
(a) Let S =
1 2 3−1 −1 5
0 4 3
. Explain why the information given in the problem assures us that S is
invertible. (You should not need to do any calculations.)
(b) Let A be the matrix of T (which is what we are looking for). Find AS. (You should be able to do this with hardly any calculation.)
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(c) Use your answer to (b) to write an expression for A (your expression may involve matrix products and inverses(1)). Then simplify A completely.
4. We’ve seen that, if a matrix A is invertible, then we can express the unique solution of A~x = ~b as ~x = A−1~b. Soon, we’ll introduce ideas that help us understand A~x = ~b better when A is not invertible. This problem is preparation for that.
Let A =
1 3 0 −3 −9 2
2 6 0 −2 −6 −5
and ~b =
2 −20
4 31
.
(a) Solve the system A~x = ~b.
(b) What does the solution set of A~x = ~b look like graphically? (Is it a line, circle, etc.? Does it pass through the origin?)
(c) Solve the system A~x = ~0. (Can you re-use your work from (a)? How does your final answer compare with your answer to (a)?)
(d) What does the solution of A~x = ~0 look like graphically? How does it relate (graphically) to the
solution set of A~x = ~b?
(e) If ~c is any vector in R4, what can you say about the number of solutions of the system A~x = ~c? (Must there be a solution? Could the system have exactly one solution? Could it have infinitely many solutions?)
5. Reflect Back (1 point).
Suppose A is an n×m matrix and ~b is a vector in Rn. Based on #4 and Problem Set 2, #1, which of the following best summarizes the relationship between the linear systems A~x = ~0 and A~x = ~b?
I. A~x = ~b must be consistent, and the solutions of A~x = ~b are exactly ~b + (the solutions of A~x = ~0).
II. A~x = ~b is not necessarily consistent, but if it is, then the solutions of A~x = ~b are exactly ~b + (the solutions of A~x = ~0).
III. A~x = ~b is not necessarily consistent, but if it is and ~x1 is one solution, then the solutions of A~x = ~b are exactly ~x1 + (the solutions of A~x = ~0).
IV. None of the above.
(1)So, for example, an answer in the form A =
[ 1 2 3 4
]−1 [ 5 6 7 8
] would be fine.
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