Find the Eigenvalues [[x,-1/3,-1/3],[-1/3,x,-1/3],[-1/3,-1/3,x]]

Math
Set up the formula to find the characteristic equation .
Substitute the known values in the formula.
Subtract the eigenvalue times the identity matrix from the original matrix.
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Multiply by each element of the matrix.
Simplify each element of the matrix .
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Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Add the corresponding elements of to each element of .
Simplify each element of the matrix .
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Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
The determinant of is .
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Set up the determinant by breaking it into smaller components.
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify each term.
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Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
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Simplify each term.
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Multiply by .
Multiply by .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by .
Subtract from .
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Move .
Subtract from .
Multiply .
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Multiply and .
Multiply by .
Expand by multiplying each term in the first expression by each term in the second expression.
Simplify each term.
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Multiply by by adding the exponents.
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Move .
Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Add and .
Multiply by by adding the exponents.
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Move .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by .
Multiply .
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Multiply by .
Multiply by .
Combine and .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
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Move .
Multiply by .
Multiply by by adding the exponents.
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Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Add and .
Combine and .
Add and .
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Move .
Add and .
Reorder the factors of .
To write as a fraction with a common denominator, multiply by .
Simplify terms.
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Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Simplify terms.
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Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Apply the distributive property.
Rewrite using the commutative property of multiplication.
Move to the left of .
Simplify each term.
Add and .
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Move .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Factor out of .
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Factor out of .
Raise to the power of .
Factor out of .
Factor out of .
Rewrite as .
Rewrite as .
Reorder and .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Simplify.
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Move to the left of .
Move to the left of .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Simplify terms.
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Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
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Simplify each term.
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Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Add and .
Add and .
Multiply by .
Subtract from .
To write as a fraction with a common denominator, multiply by .
Simplify terms.
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Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Apply the distributive property.
Multiply by .
Rewrite using the commutative property of multiplication.
Move to the left of .
Combine the numerators over the common denominator.
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify each term.
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Apply the distributive property.
Multiply .
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Multiply by .
Multiply by .
Combine and .
Combine and .
Multiply .
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Multiply and .
Multiply by .
Apply the distributive property.
Simplify.
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Multiply .
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Multiply and .
Multiply by .
Multiply .
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Multiply and .
Multiply by .
Multiply .
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Multiply and .
Multiply by .
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify each term.
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Multiply .
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Multiply by .
Multiply by .
Multiply and .
Multiply by .
Apply the distributive property.
Multiply .
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Multiply by .
Multiply by .
Apply the distributive property.
Combine and .
Multiply .
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Multiply by .
Multiply by .
Combine and .
Apply the distributive property.
Simplify.
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Multiply .
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Multiply and .
Multiply by .
Multiply .
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Multiply by .
Multiply by .
Multiply and .
Multiply by .
Multiply .
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Multiply and .
Multiply by .
Combine fractions.
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Combine fractions with similar denominators.
Simplify the expression.
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Subtract from .
Move the negative in front of the fraction.
Simplify each term.
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Move the negative in front of the fraction.
Move the negative in front of the fraction.
Simplify terms.
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Combine the numerators over the common denominator.
Add and .
Combine the numerators over the common denominator.
Subtract from .
Combine the numerators over the common denominator.
Add and .
Combine the numerators over the common denominator.
Subtract from .
Cancel the common factor of and .
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Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Apply the distributive property.
Simplify.
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Move to the left of .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Can’t combine different size matrices.
The combined expressions are .
Set the characteristic polynomial equal to to find the eigenvalues .
Solve the equation for .
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Multiply both sides of the equation by .
Remove parentheses.
Multiply by .
Find the Eigenvalues [[x,-1/3,-1/3],[-1/3,x,-1/3],[-1/3,-1/3,x]]

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