# 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 .

Simplify element by multiplying .

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.

Simplify element of the matrix.

Simplify element of the matrix.

Simplify element of the matrix.

Simplify element of the matrix.

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 .

Simplify by multiplying through.

Add and .

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 terms.

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.

Rewrite using the commutative property of multiplication.

Multiply by by adding the exponents.

Combine and

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

Multiply by .

Simplify .

Multiply by .

Multiply by .

Simplify .

Multiply by .

Multiply by .

Multiply by .

Add and .

Multiply by .

Add and .

Expand by multiplying each term in the first expression by each term in the second expression.

Simplify terms.

Remove unnecessary parentheses.

Simplify each term.

Multiply by by adding the exponents.

Move .

Combine and

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Rewrite using the commutative property of multiplication.

Multiply by by adding the exponents.

Combine and

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Simplify by adding terms.

Add and .

Add and .

Since the matrix is multiplied by , the determinant is .

Subtract from .

Simplify by adding numbers.

Add and .

Simplify by adding zeros.

Add and .

Add and .

Factor the characteristic polynomial.

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor.

Factor by grouping.

For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .

Factor out of .

Rewrite as plus

Apply the distributive property.

Multiply by .

Remove parentheses.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

Factor out the greatest common factor (GCF) from each group.

Factor the polynomial by factoring out the greatest common factor, .

Remove unnecessary parentheses.

Set the characteristic polynomial equal to to find the eigenvalues .

If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .

Set the first factor equal to .

Set the next factor equal to and solve.

Set the next factor equal to .

Add to both sides of the equation.

Multiply each term in by

Multiply each term in by .

Simplify .

Multiply by .

Multiply by .

Multiply by .

Set the next factor equal to and solve.

Set the next factor equal to .

Add to both sides of the equation.

The final solution is all the values that make true.