CHARACTERISTIC EQUATION OF MATRIX

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.

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

Example :

Determine the characteristic roots of the matrix

Solution :


Let A =  

 
0 1 2
1 0 -1
2 -1 0
 

Unit matrix of order 3x3,  I =   

 
1 0 0
0 1 0
0 0 1
 

Multiply unit matrix I by λ. 

= -λ(λ2 - 1) - 1[-λ - (-2)] + 2[-1 - (-2 λ)]

= -λ(λ- 1) - 1(-λ + 2) + 2 (-1 +2λ)

= -λ3 + λ  + λ - 2 - 2 + 4λ

= -λ3 + 2λ - 2 - 2 + 4λ

= -λ3 + 6λ - 4

To find roots, equate |A-λI| to zero. 

3 + 6λ - 4 = 0

λ3 - 6λ + 4 = 0

By trial and error, we can check the values 1 or -1 or 2 or -2...... as a root for the above equation using synthetic division. 

One of the roots is λ = 2.

To get the other two roots, solve the resulting equation λ2 + 2λ - 2 = 0 in the above synthetic division using quadratic formula. 

λ = [-b ± √(b2 -4ac)]/2a

In λ2 + 2λ - 2 = 0, a = 1, b = 2 and c = -2. 

Substitute the values of a, b and c in the quadratic formula. 

λ = [-2 ± √(4 + 8)]/2

= [-2 ± √12]/2

= [-2 ± √12]/2

= [-2 ± 2√3]/2

= -1 ± √3

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

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