Complete the following questions and I will provide the lecture notes if you need.

Linear Algebra (MATH140/MA-UY3044)

Fall 2019

Homework 8: sections 5.3, 6.1 (lectures 16, 17)

(Due Wednesday 11/13/2019 before midnight)

1. (10 points) Use cofactor expansions to compute the determinant of each of the following

matrices.

0

4 1

(a) A1 = 5 −3 0 .

2

3 1

1 2 4

(b) A2 = 3 1 1 .

2 4 2

1 −2

5 2

0

0

3 0

(c) A3 =

2 −4 −3 5 .

2

0

3 5

1 1 4

2. (10 points) Let A = 1 2 2.

1 2 5

(a) Find the cofactors of A, put them into the cofactor matrix C.

(b) Use A and C to compute det A.

(c) Use part (b) to compute A−1 .

(d) Suppose that the 4 in A was changed to 100. Which of C, det A, and A−1 would

change?

Linear Algebra (MATH140/MA-UY3044)

Fall 2019

3. (10 points) Label the following statements as true or false. Justify your choice to

receive full credit.

(a) If an n × n matrix has one eigenvector, then it has an infinite number of eigenvectors.

(b) Any n × n matrix that has fewer than n distinct eigenvalues is not diagonalizable.

(c) If λ is an eigenvalue of an invertible n × n matrix A, then λ is also an eigenvalue

of A−1 .

(d) λ = −2 is an eigenvalue of the matrix

7

3

.

3 −1

1

3 6 7

(e) −2 is an eigenvector of 3 3 7 .

1

5 6 5

4. (10 points) Compute the eigenvalues of the following

4

−3 1

(a)

0

−3 8

(c)

1

matrices.

0

0

0

0

0 −3

(b)

2 1

1 4

1

0 −1

0

(d) 1 −3

4

0

1

4

0 1

5. (10 points) Consider the matrix A = −1 −1 0 .

−2

0 1

(a) Compute the eigenvalues of A.

(b) Compute the eigenvectors associated with each of the eigenvalues in A.

(c) Use your results in (a) and (b) to diagonalize A.