Please answer part 4 of this question. Problem 5Diagonalize the following matrices:1 01. А —3 212. В —1110 23. С —2 02 04. D =12 -111

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Description: Please answer part 4 of this question. Problem 5Diagonalize the following matrices:1 01. А —3 212. В —1110 23. С —2 02 04. D =12 -111

Answered: 1 0 %3D 3 2

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Please answer part 4 of this question.

Problem 5
Diagonalize the following matrices:
1 0
1. А —
3 2
1
2. В —
1
1
1
0 2
3. С —
2 0
2 0
4. D =
1
2 -1
1
1

Expert Solution

Step 1

as you asking too many question i only answered first one.

A can be diagonalized if there exists an invertible matrix P and diagonal matrix D such that A=PDP^-1

Here A

	1	0
	3	2

Find eigenvalues of the matrix A

|A-λI|=0

	(1-λ)	0
	3	(2-λ)

= 0

∴(1-λ)×(2-λ)-0×3=0

∴(2-3λ+λ2)-0=0

∴(λ2-3λ+2)=0

∴(λ-1)(λ-2)=0

∴(λ-1)=0or(λ-2)=0

∴ The eigenvalues of the matrix A are given by λ=1,2,

1. Eigenvectors for λ=1

A-λI=

	1	0
	3	2

- 1

	1	0
	0	1

	1	0
	3	2

	1	0
	0	1

	0	0
	3	1

Now, reduce this matrix
interchanging rows R₁↔R₂

	3	1
	0	0

R₁←R₁÷3

	1	0.33333333
	0	0

The system associated with the eigenvalue λ=1

(A-1I)

	x₁
	x₂

	1	0.33333333
	0	0

	x₁
	x₂

	0
	0

⇒x₁+0.33333333x₂=0

⇒x₁=-0.33333333x₂

∴ eigenvectors corresponding to the eigenvalue λ=1 is

	-0.33333333x₂
	x₂

Let x₂=1

v1=

2. Eigenvectors for λ=2

	-0.33333333
	1

A-λI=

	1	0
	3	2

- 2

	1	0
	0	1

	1	0
	3	2

	2	0
	0	2

	-1	0
	3	0

Now, reduce this matrix
interchanging rows R₁↔R₂

	3	0
	-1	0

R₁←R₁÷3

	1	0
	-1	0

R₂←R₂+R₁

	1	0
	0	0

The system associated with the eigenvalue λ=2

(A-2I)

	x₁
	x₂

	1	0
	0	0

	x₁
	x₂

	0
	0

⇒x₁=0

∴ eigenvectors corresponding to the eigenvalue λ=2 is

	0
	x₂

Let x₂=1

v₂=

	0
	1

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