Linear Algebra Examples

Find the Eigenvalues [[3,0,0],[-3,9,18],[2,-3,-6]]
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
Add and .
Step 4.3.5
Add and .
Step 4.3.6
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
Multiply by .
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
Expand using the FOIL Method.
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Step 5.4.2.1.1.1
Apply the distributive property.
Step 5.4.2.1.1.2
Apply the distributive property.
Step 5.4.2.1.1.3
Apply the distributive property.
Step 5.4.2.1.2
Simplify and combine like terms.
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Step 5.4.2.1.2.1
Simplify each term.
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Step 5.4.2.1.2.1.1
Multiply by .
Step 5.4.2.1.2.1.2
Multiply by .
Step 5.4.2.1.2.1.3
Multiply by .
Step 5.4.2.1.2.1.4
Rewrite using the commutative property of multiplication.
Step 5.4.2.1.2.1.5
Multiply by by adding the exponents.
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Step 5.4.2.1.2.1.5.1
Move .
Step 5.4.2.1.2.1.5.2
Multiply by .
Step 5.4.2.1.2.1.6
Multiply by .
Step 5.4.2.1.2.1.7
Multiply by .
Step 5.4.2.1.2.2
Add and .
Step 5.4.2.1.3
Multiply .
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Step 5.4.2.1.3.1
Multiply by .
Step 5.4.2.1.3.2
Multiply by .
Step 5.4.2.2
Combine the opposite terms in .
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Step 5.4.2.2.1
Add and .
Step 5.4.2.2.2
Add and .
Step 5.4.2.3
Reorder and .
Step 5.5
Simplify the determinant.
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Step 5.5.1
Combine the opposite terms in .
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Step 5.5.1.1
Add and .
Step 5.5.1.2
Add and .
Step 5.5.2
Expand using the FOIL Method.
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Step 5.5.2.1
Apply the distributive property.
Step 5.5.2.2
Apply the distributive property.
Step 5.5.2.3
Apply the distributive property.
Step 5.5.3
Simplify and combine like terms.
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Step 5.5.3.1
Simplify each term.
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Step 5.5.3.1.1
Multiply by .
Step 5.5.3.1.2
Multiply by by adding the exponents.
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Step 5.5.3.1.2.1
Move .
Step 5.5.3.1.2.2
Multiply by .
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Step 5.5.3.1.2.2.1
Raise to the power of .
Step 5.5.3.1.2.2.2
Use the power rule to combine exponents.
Step 5.5.3.1.2.3
Add and .
Step 5.5.3.1.3
Rewrite using the commutative property of multiplication.
Step 5.5.3.1.4
Multiply by by adding the exponents.
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Step 5.5.3.1.4.1
Move .
Step 5.5.3.1.4.2
Multiply by .
Step 5.5.3.1.5
Multiply by .
Step 5.5.3.2
Add and .
Step 5.5.4
Move .
Step 5.5.5
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
Factor the left side of the equation.
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Step 7.1.1
Factor out of .
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Step 7.1.1.1
Factor out of .
Step 7.1.1.2
Factor out of .
Step 7.1.1.3
Factor out of .
Step 7.1.1.4
Factor out of .
Step 7.1.1.5
Factor out of .
Step 7.1.2
Factor using the perfect square rule.
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Step 7.1.2.1
Rewrite as .
Step 7.1.2.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 7.1.2.3
Rewrite the polynomial.
Step 7.1.2.4
Factor using the perfect square trinomial rule , where and .
Step 7.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 7.3
Set equal to .
Step 7.4
Set equal to and solve for .
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Step 7.4.1
Set equal to .
Step 7.4.2
Solve for .
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Step 7.4.2.1
Set the equal to .
Step 7.4.2.2
Add to both sides of the equation.
Step 7.5
The final solution is all the values that make true.