# Linear Algebra Examples

Set up the formula to find the characteristic equation .

Substitute the known values in the formula.

Multiply by each element of the matrix.

Simplify each element of the matrix .

Simplify element by multiplying .

Simplify element by multiplying .

Simplify element by multiplying .

Simplify element by multiplying .

Combine the similar matrices with each others.

Simplify each element of the matrix .

Combine the same size matrices and by adding the corresponding elements of each.

Simplify element of the matrix.

Simplify element of the matrix.

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 .

Multiply by .

Multiply by .

Multiply by .

Subtract from .

Multiply by .

Subtract from .

Set the characteristic polynomial equal to to find the eigenvalues .

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 by .

Multiply by .

Add and .

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 by .

Multiply by .

Add and .

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 .

Add and .

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.

The result can be shown in multiple forms.

Exact Form:

Decimal Form: