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 = |
|
|
Unit matrix of order 3x3, I = |
|
Multiply unit matrix I by λ.
= -λ(λ2 - 1) - 1[-λ - (-2)] + 2[-1 - (-2 λ)]
= -λ(λ2 - 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.
-λ3 + 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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