Question

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

Answer 1

Answer:

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]