Finite Math Examples

Find the Eigenvalues [[9,1,0],[3/2,0,-2],[1,2,4]]
Step 1
Set up the formula to find the characteristic equation .
Step 2
The identity matrix or unit matrix of size is the square matrix with ones on the main diagonal and zeros elsewhere.
Step 3
Substitute the known values into .
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Step 3.1
Substitute for .
Step 3.2
Substitute for .
Step 4
Simplify.
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Step 4.1
Simplify each term.
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Step 4.1.1
Multiply by each element of the matrix.
Step 4.1.2
Simplify each element in the matrix.
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Step 4.1.2.1
Multiply by .
Step 4.1.2.2
Multiply .
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Step 4.1.2.2.1
Multiply by .
Step 4.1.2.2.2
Multiply by .
Step 4.1.2.3
Multiply .
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Step 4.1.2.3.1
Multiply by .
Step 4.1.2.3.2
Multiply by .
Step 4.1.2.4
Multiply .
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Step 4.1.2.4.1
Multiply by .
Step 4.1.2.4.2
Multiply by .
Step 4.1.2.5
Multiply by .
Step 4.1.2.6
Multiply .
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Step 4.1.2.6.1
Multiply by .
Step 4.1.2.6.2
Multiply by .
Step 4.1.2.7
Multiply .
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Step 4.1.2.7.1
Multiply by .
Step 4.1.2.7.2
Multiply by .
Step 4.1.2.8
Multiply .
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Step 4.1.2.8.1
Multiply by .
Step 4.1.2.8.2
Multiply by .
Step 4.1.2.9
Multiply by .
Step 4.2
Add the corresponding elements.
Step 4.3
Simplify each element.
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Step 4.3.1
Add and .
Step 4.3.2
Add and .
Step 4.3.3
Add and .
Step 4.3.4
Subtract from .
Step 4.3.5
Add and .
Step 4.3.6
Add and .
Step 4.3.7
Add and .
Step 5
Find the determinant.
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Step 5.1
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.
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Step 5.1.1
Consider the corresponding sign chart.
Step 5.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 5.1.3
The minor for is the determinant with row and column deleted.
Step 5.1.4
Multiply element by its cofactor.
Step 5.1.5
The minor for is the determinant with row and column deleted.
Step 5.1.6
Multiply element by its cofactor.
Step 5.1.7
The minor for is the determinant with row and column deleted.
Step 5.1.8
Multiply element by its cofactor.
Step 5.1.9
Add the terms together.
Step 5.2
Multiply by .
Step 5.3
Evaluate .
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Step 5.3.1
The determinant of a matrix can be found using the formula .
Step 5.3.2
Simplify the determinant.
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Step 5.3.2.1
Simplify each term.
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Step 5.3.2.1.1
Apply the distributive property.
Step 5.3.2.1.2
Multiply by .
Step 5.3.2.1.3
Rewrite using the commutative property of multiplication.
Step 5.3.2.1.4
Simplify each term.
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Step 5.3.2.1.4.1
Multiply by by adding the exponents.
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Step 5.3.2.1.4.1.1
Move .
Step 5.3.2.1.4.1.2
Multiply by .
Step 5.3.2.1.4.2
Multiply by .
Step 5.3.2.1.4.3
Multiply by .
Step 5.3.2.1.5
Multiply by .
Step 5.3.2.2
Reorder and .
Step 5.4
Evaluate .
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Step 5.4.1
The determinant of a matrix can be found using the formula .
Step 5.4.2
Simplify the determinant.
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Step 5.4.2.1
Simplify each term.
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Step 5.4.2.1.1
Apply the distributive property.
Step 5.4.2.1.2
Cancel the common factor of .
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Step 5.4.2.1.2.1
Factor out of .
Step 5.4.2.1.2.2
Cancel the common factor.
Step 5.4.2.1.2.3
Rewrite the expression.
Step 5.4.2.1.3
Multiply by .
Step 5.4.2.1.4
Combine and .
Step 5.4.2.1.5
Multiply by .
Step 5.4.2.2
Add and .
Step 5.5
Simplify the determinant.
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Step 5.5.1
Add and .
Step 5.5.2
Simplify each term.
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Step 5.5.2.1
Expand by multiplying each term in the first expression by each term in the second expression.
Step 5.5.2.2
Simplify each term.
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Step 5.5.2.2.1
Multiply by .
Step 5.5.2.2.2
Multiply by .
Step 5.5.2.2.3
Multiply by by adding the exponents.
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Step 5.5.2.2.3.1
Move .
Step 5.5.2.2.3.2
Multiply by .
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Step 5.5.2.2.3.2.1
Raise to the power of .
Step 5.5.2.2.3.2.2
Use the power rule to combine exponents.
Step 5.5.2.2.3.3
Add and .
Step 5.5.2.2.4
Rewrite using the commutative property of multiplication.
Step 5.5.2.2.5
Multiply by by adding the exponents.
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Step 5.5.2.2.5.1
Move .
Step 5.5.2.2.5.2
Multiply by .
Step 5.5.2.2.6
Multiply by .
Step 5.5.2.2.7
Multiply by .
Step 5.5.2.3
Add and .
Step 5.5.2.4
Subtract from .
Step 5.5.2.5
Apply the distributive property.
Step 5.5.2.6
Multiply .
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Step 5.5.2.6.1
Multiply by .
Step 5.5.2.6.2
Multiply by .
Step 5.5.2.7
Multiply by .
Step 5.5.3
To write as a fraction with a common denominator, multiply by .
Step 5.5.4
Combine and .
Step 5.5.5
Combine the numerators over the common denominator.
Step 5.5.6
Simplify each term.
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Step 5.5.6.1
Simplify the numerator.
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Step 5.5.6.1.1
Factor out of .
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Step 5.5.6.1.1.1
Factor out of .
Step 5.5.6.1.1.2
Factor out of .
Step 5.5.6.1.1.3
Factor out of .
Step 5.5.6.1.2
Multiply by .
Step 5.5.6.1.3
Add and .
Step 5.5.6.2
Move to the left of .
Step 5.5.6.3
Move the negative in front of the fraction.
Step 5.5.7
Subtract from .
Step 5.5.8
Reorder and .
Step 6
Set the characteristic polynomial equal to to find the eigenvalues .
Step 7
Solve for .
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Step 7.1
Graph each side of the equation. The solution is the x-value of the point of intersection.