# 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 .

Apply the distributive property.

Simplify .

Multiply by .

Multiply by .

Multiply by .

Simplify the expression.

Remove unnecessary parentheses.

Add and .

Since the matrix is multiplied by , the determinant is .

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.

Multiply by by adding the exponents.

Move .

Combine and

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Simplify .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Subtract from .

Multiply by .

Subtract from .

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 .

Multiply by by adding the exponents.

Move .

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 .

Add and .

Add and .

Subtract from .

Set the characteristic polynomial equal to to find the eigenvalues .

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