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 :

Now we have to multiply λ with unit matrix I.

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