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# Assignment 5 Inverses

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Assignment 5
Inverses
1. Find the inverse of the following matrices or show that the inverse does not exist:
a)

1 1 2
1 2 2
1 −1 3

 b)

2 3 −1
4 2 1
−2 5 −5

2. Consider the following system of equations
x1 + 2×2 + 2×3 = b1
x2 + 2×3 = b2
−x1 + 3×2 + 12×3 = b3.
By finding the inverse of the coefficient matrix associated with the system of equations, find an
expression for the solution ~x in terms of arbitrary b1, b2, and b3.
3. Suppose A is an n × n invertible matrix and your friend claims to have found two different matrices
B and C that work as inverses. Prove that your friend is mistaken – that is, prove that there is only
one inverse of A. (Hint: Assume your friend’s claim is true but then show that this implies B = C.)
4. Suppose A and B are n × n matrices such that AB = I. Must it be true that rank(A) = rank(B)?
5. If {~v1, ~v2, . . . , ~vk} is a linearly independent set of vectors in R
n and A is an n × n invertible matrix.
Prove that the set {A~v1, A~v2, . . . , A~vk} is also linearly independent. (Hint: Assume that the latter
set is linearly dependent and show that this gives rise to a contradiction.)
Determinants
6. Determine if the following matrices are invertible by finding the determinant (you do not need to
find the inverse):
a)

1 5 −3
2 13 −7
3 −3 3

 b)

2 0 0 8
1 −7 −5 0
3 8 6 0
0 7 5 4

c)

1 0 0 1 1
1 1 0 0 1
0 1 1 1 1
0 0 0 1 0
0 1 0 1 0

7. For what values of s is the matrix

1 s 1
s −3 −2s
1 2 −1

 invertible?
8. It turns out all of our results regarding inverses apply equally well to complex-valued matrices. This
includes using the determinant to check if a matrix is invertible. With this in mind, determine if
the following matrix is invertible:
?
1 + i 1 − i
1 + 3i 3 − i
?
.
1
9. In the previous assignment we defined the transpose of a matrix A – denoted AT
– by (AT
)ij = (A)ji.
Our goal now is to argue that, for an arbitrary n × n matrix we have det(AT
) = det(A). Consider
the cofactor expansion of an arbitrary 3 × 3 matrix A along its first column and compare it to the
cofactor expansion of AT along its first row. Are the determinants the same? Will this argument
10. In tutorial, we established that determinants obey the property that for any two n × n matrices A
and B, we have det(AB) = det(A)det(B). Use this result to solve the following questions:
(a) Consider three matrices A, B, and C. Given that det(AB) = 6, det(BC) = 12, and
det(ABC) = 24, find the det(B).
(b) Suppose a matrix A satisfies A3 = A. What are the possible values for the determinant of A?
(c) In assignment 4 we defined an orthogonal matrix as one that satisfies AT A = I. What values
can the determinant of an orthogonal matrix have? (Hint: Use the result of question 9.)
(d) When we study diagonalization, we will introduce the notion of similar matrices. In particular,
if A and B are similar then there an invertible matrix P such that B = P
−1AP. In this case,
show that det(A) = det(B).
Eigenvalues and Eigenvectors
11. The matrix A =
?
2 0
0 3 ?
performs an unequal scaling on an input vector. In particular, it doubles
the x1 component and triples the x2 component.
(a) Consider the following three vectors: ~v1 =
?
1
0
?
, ~v2 =
?
0
1
?
, and ~v3 =
?
1
1
?
. Compute A~v1, A~v2,
and A~v3.
(b) On three separate graphs, sketch each input along with its output (e.g., on one graph, sketch
~v1 and A~v1).
(c) Which of the vectors ~v1, ~v2, and ~v3 are eigenvectors? What are their associated eigenvalues?
12. Find the eigenvalues and associated eigenvectors for the following matrices:
a) ?
2 1
1 2 ?
b)

1 1 −1
0 0 2
0 0 2

2 Assignment 5 Inverses
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