Description
The question is in the attached .pdf file.
I am going to need comprehensible solutions (if possible step-by-step), please.
Unformatted Attachment Preview
Purchase answer to see full attachment
Explanation & Answer
The solutions are ready. I believe there is a quicker solution for c) but y process is correct, too. Please ask if something is unclear.docx and pdf files are identical
0
0 −2 0
0
0
1
0
𝐴= 1
1
2
0
−1 −1 −2 −1
1
2
1
(1
−𝜆 0
−2
0 −𝜆
1
a1) Eigenvalues. det(𝐴 − 𝜆𝐼) = || 1
1 2−𝜆
−1 −1 −2
1
1
2
0
0
0
−1 − 𝜆
1
0
0
0 .
−2
2)
0
0
0 ||.
−2
2−𝜆
We will open all determinants using their first rows:
−𝜆
1
1 2−𝜆
det(𝐴 − 𝜆𝐼) = −𝜆 |
−1 −2
1
2
2−𝜆
𝜆 | −2
2
2
0
−1 − 𝜆
1
0
0 −𝜆
0
0
1
1
0
|− 2|
−2
−1 −1 −1 − 𝜆
2−𝜆
1
1
1
1
0
0
−2 | + 𝜆 |−1 −1 − 𝜆
1
1
2−𝜆
−1 − 𝜆
= 𝜆2 (2 − 𝜆) |
1
−1 − 𝜆
= (𝜆2 (2 − 𝜆) − 𝜆) |
1
0
0
−1 − 𝜆
1
−2
−1 − 𝜆
|+𝜆|
2−𝜆
1
0
1
0
−2 | − 2𝜆 |−1 −1 − 𝜆
2−𝜆
1
1
−2
−1 − 𝜆
| − 2𝜆 |
2−𝜆
1
0
0
|=
−2
2−𝜆
0
−2 | =
2−𝜆
−2
|=
2−𝜆
−2
| = 𝜆(2𝜆 − 𝜆2 − 1)((𝜆 + 1)(𝜆 − 2) + 2) = −𝜆(𝜆 − 1)2 𝜆(𝜆 − 1) = −𝜆2 (𝜆 − 1)3 .
2−𝜆
Thus the eigenvalues are 𝝀𝟏 = 𝟎 and 𝝀𝟐 = 𝟏.
a2) Eigenspaces. Consider the equations 𝐴𝑥 = 0, 𝐴𝑥 = 𝑥. Note that the first and the second, the fourth and the fifth rows of 𝐴 are
proportional.
0
0
𝐴𝑥 = 0,
1
−1
(1
𝑥1
−2𝑥3 = 0
0 −2 0
0
𝑥
𝑥3 = 0
𝑥3 = 0
𝑥3 = 0
2
0
1
0
0
𝑥3 = 0,
𝑥1 + 𝑥2 + 2𝑥3 = 0
𝑥1 + 𝑥2 + 2𝑥3 = 0
⇔ {
⇔ { 𝑥1 + 𝑥2 = 0 .
1
2
0
0
𝑥
−𝑥1 − 𝑥2 − 2𝑥3 − 𝑥4 − 2𝑥5 = 0
𝑥1 + 𝑥2 + 2𝑥3 + 𝑥4 + 2𝑥5 = 0
𝑥4 + 2𝑥5 = 0
−1 −2 −1 −2
4
{ 𝑥1 + 𝑥2 + 2𝑥3 + 𝑥4 + 2𝑥5 = 0
1
2
1
2 ) (𝑥5 )
𝟏
𝟎
𝟎
−𝟏
Therefore the subspace 𝐾𝑒𝑟(𝐴) is 2-dimensional (3 independent constraints) and has a basis
.
𝟎 ,
𝟎
−𝟐
𝟎
{( 𝟏 ) ( 𝟎 )}
−2𝑥3 = 𝑥1
𝑥1 = −2𝑥3
𝑥1 = 0
𝑥1 = −2𝑥3
𝑥1 = −2𝑥3
𝑥3 = 𝑥2
𝑥2 = 𝑥3
𝑥2 = 𝑥3
𝑥2 = 𝑥3
𝑥2 = 0
𝑥1 + 𝑥2 + 2𝑥3 = 𝑥3
𝑥3 = 𝑥3
𝐴𝑥 = 𝑥,
⇔
⇔ {
⇔ {
⇔ {
.
𝑥5 = −𝑥4
𝑥5 = −𝑥4
𝑥5 = −𝑥4
𝑥5 = −𝑥4
−𝑥1 − 𝑥2 − 2𝑥3 − 𝑥4 − 2𝑥5 = 𝑥4
𝑥3 + 𝑥4 + 𝑥5 = 0
𝑥3 = 0
𝑥3 = 0
{ 𝑥1 + 𝑥2 + 2𝑥3 + 𝑥4 + 2𝑥5 = 𝑥5
{𝑥3 + 𝑥4 + 2𝑥5 = 𝑥5
𝟎
𝟎
T...
Review
Review
24/7 Homework Help
Stuck on a homework question? Our verified tutors can answer all questions, from basic math to advanced rocket science!
Similar Content
Related Tags
Extreme Ownership - How US Navy SEALs Lead and Win
by Jocko Willink and Leif Babin
The Girl With The Dragon Tattoo
by Stieg Larsson
Heart of Darkness
by Joseph Conrad
Robinson Crusoe
by Daniel Defoe
The Red Badge of Courage
by Stephen Crane
Ezperanza Rising
by Pam Muñoz Ryan
Catch-22
by Joseph Heller
50 Shades of Grey
by E. L. James
The Picture of Dorian Gray
by Oscar Wilde