Characteristic equation of matrix :
Here 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 AλI is called characteristic polynomial of matrix.
Then the equation AλI = 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

Solution :
Let A = 

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

Now we have to multiply λ with unit matrix I.
λI = 

AλI= 

 

= 

= 

AλI= 
 
= 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 AλI = 0
λ³ + 6λ  4 = 0
λ³  6λ + 4 = 0
For solving this equation first let us do synthetic division.
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 ± √2²4(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.
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