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
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Solution :
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Let A = |
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The order of A is 3 x 3. So the unit matrix I = |
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Now we have to multiply λ with unit matrix I.
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λI = |
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| A-λI= |
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- |
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| = |
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| A-λI= |
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= -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. 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.
