Find the eigenvectors and eigenvalues for the matrix. A =

[2 - 2

-4 4]

Consider lambda as $

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

eigen vectors:

[1 2]

[0 0]

Step-by-step 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]