Characteristic Equation of Matrix





In this page characteristic equation of matrix we are going to see how to find characteristic equation of any matrix with detailed example.

Definition :

Let A be any square matrix of order n x n and I be a unit matrix of same order. Then |A-λI| is called characteristic polynomial of matrix. 

Then the equation |A-λI| = 0 is called characteristic roots of matrix.  The roots of this equation is called characteristic roots of matrix.

Another name of characteristic roots:

characteristic roots are also known as latent roots or eigenvalues of a matrix.

Example :

Determine the characteristic roots of the matrix

 
0 1 2
1 0 -1
2 -1 0
 


Solution :


   Let A =

 
0 1 2
1 0 -1
2 -1 0
 

The order of A is 3 x 3. So the unit matrix I =

 
1 0 0
0 1 0
0 0 1
 

Now we have to multiply λ with unit matrix I.

  λI =

 
λ 0 0
0 λ 0
0 0 λ
 
A-λI=
 
0 1 2
1 0 -1
2 -1 0
 
-
 
λ 0 0
0 λ 0
0 0 λ
 
 
                      
  =
 
(0-λ)   (1-0)   (2-0)
(1-0)   (0-λ)   (-1-0)
(2-0)   (-1-0)   (0-λ)
 
 
  =
 
  1   2
1     -1
2   -1  
 
 
A-λI=
 
  1   2
1     -1
2   -1  
 

  =  -λ( λ ² - 1) - 1 (-λ - (-2) ) + 2 (-1 - (-2 λ) )

  =  -λ( λ ² - 1) - 1 (-λ + 2) ) + 2 (-1 +2 λ)

  =  -λ³ + λ  + λ - 2 - 2 + 4 λ

  =  -λ³ + 2λ  - 2 - 2 + 4 λ

  =  -λ³ + 6λ - 4 

To find roots let |A-λI| = 0

              -λ³ + 6λ - 4 = 0

              λ³ - 6λ + 4 = 0

For solving this equation first let us do synthetic division.  characteristic equation of matrix

By using synthetic division we have found one value of λ that is λ = 2.

Now we have to solve λ² + 2 λ - 2 to get another two values. For that we have to use quadratic formula (-b ± √b² -4ac)/2a

a = 1 b = 2 and c = -2

           x = [-2 ± √2²-4(1)(-2)]/2(1)

           x = [-2 ± √4+8]/2(1)

           x = [-2 ± √12]/2

           x = [-2 ± 2√3]/2

           x = 2[-1 ± √3]/2

           x = -1 ± √3

Therefore the characteristic roots are x = 1, -1 ± √3.






Characteristic Equation of Matrix to Rank method
HTML Comment Box is loading comments...
Enjoy this page? Please pay it forward. Here's how...

Would you prefer to share this page with others by linking to it?

  1. Click on the HTML link code below.
  2. Copy and paste it, adding a note of your own, into your blog, a Web page, forums, a blog comment, your Facebook account, or anywhere that someone would find this page valuable.












Featured Categories 

Math Word Problems

SAT Math Worksheet

P-SAT Preparation

Math Calculators

Quantitative Aptitude

Transformations

Algebraic Identities

Trig. Identities

SOHCAHTOA

Multiplication Tricks

PEMDAS Rule

Types of Angles

Aptitude Test