Characteristic equation of matrix :
Here we are going to see how to find characteristic equation of any matrix with detailed example.
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.
Determine the characteristic roots of the matrix
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.
= -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 |A-λI| = 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 ± √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.
After having gone through the stuff given above, we hope that the students would have understood "Characteristic equation of matrix"
Apart from the stuff given above, if you want to know more about "Characteristic equation of matrix", please click here.
Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
If you have any feedback about our math content, please mail us :
We always appreciate your feedback.
You can also visit the following web pages on different stuff in math.
APTITUDE TESTS ONLINE
ACT MATH ONLINE TEST
TRANSFORMATIONS OF FUNCTIONS
ORDER OF OPERATIONS
Analytical geometry calculators
MATH FOR KIDS
Word problems on linear equations
Trigonometry word problems
Word problems on mixed fractrions
Ratio and proportion shortcuts
Converting repeating decimals in to fractions