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.
Characteristic roots are also known as latent roots or eigenvalues of a matrix.
Example :
Determine the characteristic roots of the matrix
Solution :
Let A = 

Unit matrix of order 3x3, I = 

Multiply unit matrix I by λ.
= λ(λ^{2}  1)  1[λ  (2)] + 2[1  (2 λ)]
= λ(λ^{2 } 1)  1(λ + 2) + 2 (1 +2λ)
= λ^{3} + λ + λ  2  2 + 4λ
= λ^{3} + 2λ  2  2 + 4λ
= λ^{3} + 6λ  4
To find roots, equate AλI to zero.
λ^{3} + 6λ  4 = 0
λ^{3}  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 λ^{2} + 2λ  2 = 0 in the above synthetic division using quadratic formula.
λ = [b ± √(b^{2} 4ac)]/2a
In λ^{2} + 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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