# Linear Algebra Examples

Step 1
Set up the formula to find the characteristic equation .
Step 2
Substitute the known values in the formula.
Step 3
Subtract the eigenvalue times the identity matrix from the original matrix.
Multiply by each element of the matrix.
Simplify each element in the matrix.
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Simplify each element of the matrix .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Step 4
Find the determinant of .
Set up the determinant by breaking it into smaller components.
Find the determinant of .
The determinant of a matrix can be found using the formula .
Simplify the determinant.
Simplify each term.
Multiply by .
Multiply by .
Simplify by multiplying through.
Apply the distributive property.
Multiply.
Multiply by .
Multiply by .
Find the determinant of .
The determinant of a matrix can be found using the formula .
Simplify the determinant.
Simplify terms.
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 .
Multiply by .
Multiply by .
Multiply .
Multiply by .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Move .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Expand by multiplying each term in the first expression by each term in the second expression.
Simplify terms.
Simplify each term.
Multiply by .
Multiply by .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Move .
Multiply by .
Multiply by .
Multiply by by adding the exponents.
Move .
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Multiply by .
Subtract from .
Subtract from .
Since the matrix is multiplied by , the determinant is .
Combine the opposite terms in .
Subtract from .
Factor the characteristic polynomial.
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Let . Substitute for all occurrences of .
Factor by grouping.
Reorder terms.
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Factor out of .
Rewrite as plus
Apply the distributive property.
Multiply by .
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
Factor out the greatest common factor (GCF) from each group.
Factor the polynomial by factoring out the greatest common factor, .
Factor.
Replace all occurrences of with .
Remove unnecessary parentheses.
Step 5
Set the characteristic polynomial equal to to find the eigenvalues .
Step 6
Find the roots of by solving for .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set equal to .
Set equal to and solve for .
Set equal to .
Solve for .
Subtract from both sides of the equation.
Divide each term in by and simplify.
Divide each term in by .
Simplify the left side.
Dividing two negative values results in a positive value.
Divide by .
Simplify the right side.
Divide by .
Set equal to and solve for .
Set equal to .
Subtract from both sides of the equation.
The final solution is all the values that make true.
Step 7