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

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.
Multiply by each element of the matrix.
Simplify each element of the matrix .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Add the corresponding elements of to each element of .
Simplify each element of the matrix .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
The determinant of is .
Set up the determinant by breaking it into smaller components.
The determinant of is .
The determinant of a matrix can be found using the formula .
Simplify the determinant.
Simplify each term.
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Multiply by .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by .
Subtract from .
Move .
Subtract from .
Multiply .
Multiply and .
Multiply by .
Expand by multiplying each term in the first expression by each term in the second expression.
Simplify each term.
Multiply by by adding the exponents.
Move .
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Multiply by by adding the exponents.
Move .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by .
Multiply .
Multiply by .
Multiply by .
Combine and .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Move .
Multiply by .
Multiply by by adding the exponents.
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Combine and .
Move .
Reorder the factors of .
To write as a fraction with a common denominator, multiply by .
Simplify terms.
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Simplify terms.
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Apply the distributive property.
Rewrite using the commutative property of multiplication.
Move to the left of .
Simplify each term.
Move .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Factor out of .
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.
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.
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Subtract from .
To write as a fraction with a common denominator, multiply by .
Simplify terms.
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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 .
The determinant of a matrix can be found using the formula .
Simplify the determinant.
Simplify each term.
Apply the distributive property.
Multiply .
Multiply by .
Multiply by .
Combine and .
Combine and .
Multiply .
Multiply and .
Multiply by .
Apply the distributive property.
Simplify.
Multiply .
Multiply and .
Multiply by .
Multiply .
Multiply and .
Multiply by .
Multiply .
Multiply and .
Multiply by .
The determinant of is .
The determinant of a matrix can be found using the formula .
Simplify the determinant.
Simplify each term.
Multiply .
Multiply by .
Multiply by .
Multiply and .
Multiply by .
Apply the distributive property.
Multiply .
Multiply by .
Multiply by .
Apply the distributive property.
Combine and .
Multiply .
Multiply by .
Multiply by .
Combine and .
Apply the distributive property.
Simplify.
Multiply .
Multiply and .
Multiply by .
Multiply .
Multiply by .
Multiply by .
Multiply and .
Multiply by .
Multiply .
Multiply and .
Multiply by .
Combine fractions.
Combine fractions with similar denominators.
Simplify the expression.
Subtract from .
Move the negative in front of the fraction.
Simplify each term.
Move the negative in front of the fraction.
Move the negative in front of the fraction.
Simplify terms.
Combine the numerators over the common denominator.
Combine the numerators over the common denominator.
Subtract from .
Combine the numerators over the common denominator.
Combine the numerators over the common denominator.
Subtract from .
Cancel the common factor of and .
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.
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 .
Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Apply the distributive property.
Simplify.
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 .
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]]