Numerical Methods and Optimization

Elements of Numerical Linear Algebra 85

(i) 2x

1

+ x

2

=1

x

1

− x

2

=2

x

1

+2x

2

=1

(ii) x

1

+ x

2

=1

2x

1

+3x

2

=0

x

1

+2x

2

=2

(iii) −x

1

+ x

2

=1

x

1

+2x

2

=0

2x

1

+ x

2

=1

(iv) −3x

1

+ x

2

= −2

4x

1

+7x

2

=11

x

1

− x

2

=3

(v) 2x

1

− x

2

+ x

3

=1

−x

1

+ x

2

− x

3

=2

x

1

+ x

2

+2x

3

= −1

2x

1

+ x

2

+2x

3

=3.

3.10. Consider the following two matrices:

A =



101 −90

110 −98



,B=



100.999 −90.001

110 −98



.

Find the eigenvalues of A and B. What do you observe?

3.11. Given the matrix A =

⎡

⎣

401

−210

−201

⎤

⎦

,

(a) Find the eigenvalues and corresponding eigenvectors of A.

(b) Use your answer from (a) to compute det(A).

(c) What are the eigenvalues of A

−1

?

3.12. Let a polynomial p(x)=c

0

+ c

1

x + c

2

x

2

+ ...+ c

n−1

x

n−1

+ x

n

be given.

Prove that p(x) is the characteristic polynomial of matrix A =[a

ij

]

n×n

defined as follows:

a

i,j

=

⎧

⎨

⎩

1, if j = i +1;

−c

j−1

if i = n,