Question

Find the eigenvectors and eigenvalues for the matrix. A =
[2 -2
-4 4]

Answer 1

Consider lambda as $

$ =0, & $ =6 (eigen values)

eigen vectors :

[1 2]

[0 0]

Explanation:

1st step:

Lambda I

here I is identity matrix.

As our matrix is 2X2 , so we will take I= [ 1 0, 0 1]

consider lambda is represented as $

so $I matrix will be [$ 0, 0 $]

2nd Step:

A - Lambda I

i.e .[2 -2, -4 4] - [$ 0, 0 $]

=> [2-$ -2, -4 4-$]

3rd Step:

Det [A-$I]

i.e. [(2-$)(4-$) - (-2)(-4)] = 0

=>8-2$-4$+$²-8

=>$²-6$

4th Step:

Det [A-$I]=0

i.e. $²-6$ =0

$($-6)=0

i.e. $ =0, & $ =6 (eigen values)

5th Step

put these eigen values in [A-$I] matrix

i.e if we put $ =0

we get [2 -2, -4 4]

consider this matrix as B

then

B X⁻= 0⁻ (⁻ is a bar sign notation)

[2 -2, -4 4] [x₁ x₂] =[0 0]

by row reduction

-2R₁+ R₂ -> R₂

so

2X₁-2X₂=0

X₁=X₂

ie eigen vector will be [0 0]

now consider

i.e if we put $ =6

we get [-4 -2, -4 -2]

consider this matrix as B

then

B X⁻= 0⁻ (⁻ is a bar sign notation)

[-4 -2, -4 -2] [x₁ x₂] =[0 0]

by row reduction

R₁- R₂ -> R₂

so

-4X₁-2X₂=0

2X₁=X₂

if we put X₁=1

x₂=2

So eigen vector will be

[1 2]