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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)?
Justify your answer.
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
generalize to an n × n matrix? Justify your answers.
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