# Find the Eigenvalues [[two , one],[three , two]]

Set up the formula to find the characteristic equation .

Substitute the known values in the formula.

First, multiply by each element of the matrix.

Clarify each element of the matrix .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Add the corresponding elements of to each element of .

Simplify each element of the matrix .

Simplify .

Simplify .

These are both valid notations for the determinant of a matrix.

The determinant of a matrix can be found using the formula .

Clarify the determinant.

Clarify each term.

Expand using the FOIL Method.

Use the distributive law.

Apply the distributive property.

Apply the distributive property.

Clarify and Add like terms.

Simplify each term.

Multiply by .

Multiply by .

Multiply by .

Initially, multiply by .

Multiply by .

Multiply by .

Subtract from .

Multiply by .

Subtract from .

Set the characteristic polynomial equal to to find the eigenvalues .

Apply the quadratic formula to solve the equation.

Substitute the values , , and into the quadratic formula and solve for .

Simplify.

Clarify the numerator.

Raise to the power of .

First, multiply by .

Multiply by .

Subtract from .

Should be rewritten as .

Factor out of .

Rewrite as .

Withdraw terms from under the radical.

Multiply by .

Simplify .

Clarify the expression to solve for the portion of the .

Simplify the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Subtract from .

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 by .

Multiply by .

Subtract from .

Rewrite as .

Factor out of .

Rewrite as .

Pull terms out from under the radical.

Multiply by .

Simplify .

Change the to .

The combination of 2 solutions is the answer.

The result can be displayed in a variety of ways.

Exact Form:

Decimal Form:

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