# 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 to get .

Simplify element by multiplying to get .

Simplify element by multiplying to get .

Simplify element by multiplying to get .

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.

Remove parentheses.

Simplify and combine like terms.

Simplify each term.

Move .

Use the power rule to combine exponents.

Add and to get .

Simplify .

Multiply by to get .

Multiply by to get .

Multiply by to get .

Multiply by to get .

Multiply by to get .

Subtract from to get .

Multiply by to get .

Subtract from to get .

Set the characteristic polynomial equal to to find the eigenvalues .

Since does not contain the variable to solve for, move it to the right side of the equation by adding to both sides.

Move to the left side of the equation by subtracting it from both sides.

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 to get .

Multiply by to get .

Multiply by to get .

Add and to get .

Rewrite as .

Pull terms out from under the radical.

Simplify the denominator.

Rewrite.

Multiply by to get .

Simplify .

The final answer is the combination of both solutions.