MATH140 Pace University New York Linear Algebra Homework 8

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.