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

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