Help with Engineering maths pls!

[ldlian987]

Hi I need some help with engineering maths here, not very good at these! Any help is greatly appreciated! Thanks!

2. (a) Find (i) the trace and (ii) the determinant of the matrix

A= 5 1 1
1 3 1
1 1 3

Given that one eigenvalue of A is 6, using results (i) and (ii), or otherwise, deduce
all the eigenvalues of A. [3]
Find the eigenvectors of A and write down a matrix P such that P(^−1)AP is a diagonal matrix

3. 3. (a) Let
f(t) = {3 , 0 ≤ t < 0.5
{0 , 0.5 ≤ t < 1

f(t + 1) = f(t)
Sketch the function in the interval −2 ≤ t ≤ 2. Find the Fourier series representation
of f(t), writing out explicitly the first five non-zero terms in the series.

 

[maturestudent]

tr(A) = 5 + 3 + 3 = 11

 

[maturestudent]

|A| = 5(9-1) - 1(3-1) + 1(1-3)
= 40-2-2
= 36

 

[maturestudent]

Trace is the sum of eigenvalues

Determinant is product of eigenvalues.

Given one of the eigenvalue is 6, let x and y be the other two eigenvalues.

6 + x + y = 11
6xy = 36 => xy = 6

6x + x^2 + xy = 11x
x^2 - 5x +6 = 0
(x-3)(x-2)=0

hence if x=2; y=3
if x =3, y=3

Eigenvalues are 6, 2, 3

 

[maturestudent]

5 1 1
A= 1 3 1
1 1 3

Eigenvalue = 6

-1 1 1
A-6I = 1 -3 1
1 1 -3

Augmented matrix for (A-6I)x = 0

-1 1 1 0
1 -3 1 0
1 1 -3 0


1 1 -3 0
~ 0 1 -1 0
0 0 0 0

x1 + x2 - 3x3 = 0
x2 - x3 =0

Eigenvector:

2
=> x = 1
1

 

[ldlian987]

5 1 1
A= 1 3 1
1 1 3

Eigenvalue = 6

-1 1 1
A-6I = 1 -3 1
1 1 -3

Augmented matrix for (A-6I)x = 0

-1 1 1 0
1 -3 1 0
1 1 -3 0


1 1 -3 0
~ 0 1 -1 0
0 0 0 0

x1 + x2 - 3x3 = 0
x2 - x3 =0

Eigenvector:

2
=> x = 1
1

thank you very much!

 

[seianko]

10 years...i rtn everything back to sch