# 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 .
Simplify each element of the matrix .
Simplify .
Simplify .
Step 4
Find the determinant of .
These are both valid notations for the determinant of a matrix.
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 by adding the exponents.
Move .
Multiply by .
Multiply by .
Multiply by .
Subtract from .
Multiply by .
Subtract from .
Step 5
Reorder the polynomial.
Step 6
Set the characteristic polynomial equal to to find the eigenvalues .
Step 7
Find the roots of by solving for .
Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
Simplify the numerator.
Raise to the power of .
Multiply .
Multiply by .
Multiply by .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply .
Multiply by .
Multiply by .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Change the to .
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply .
Multiply by .
Multiply by .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Change the to .
The final answer is the combination of both solutions.
Step 8
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 9