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 .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Add the corresponding elements of to each element of .

Simplify each element of the matrix .

Simplify .

Simplify .

Simplify .

Simplify .

Simplify .

Simplify .

Simplify .

Simplify .

Simplify .

Set up the determinant by breaking it into smaller components.

The determinant of is .

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

Simplify the determinant.

Simplify each term.

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Apply the distributive property.

Multiply by by adding the exponents.

Move .

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

The determinant of is .

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 .

The determinant of is .

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

Simplify the determinant.

Simplify each term.

Multiply by .

Move to the left of .

Simplify by multiplying through.

Apply the distributive property.

Multiply.

Multiply by .

Multiply by .

Add and .

Add and .

Add and .

Can’t combine different size matrices.

The combined expressions are .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Set the characteristic polynomial equal to to find the eigenvalues .

Simplify .

Apply the distributive property.

Simplify.

Multiply by .

Multiply by .

Multiply by .

Rewrite as .

Graph each side of the equation. The solution is the x-value of the point of intersection.

Find the Eigenvalues [[0,0.1,0.9],[0.6,0,0.4],[0.9,0.1,0]]