# 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 .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

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 .

Simplify .

Simplify .

Simplify .

Set up the determinant by breaking it into smaller components.

The determinant of is .

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 terms.

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 .

Add and .

Multiply by .

Combine the opposite terms in .

Add and .

Add and .

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 by adding the exponents.

Move .

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Add and .

The determinant of is .

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

Simplify the determinant.

Simplify each term.

Apply the distributive property.

Multiply by .

Multiply by .

Multiply by .

Simplify by multiplying through.

Subtract from .

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 .

Apply the distributive property.

Multiply .

Multiply by .

Multiply by .

Multiply by .

Apply the distributive property.

Move to the left of .

Multiply by .

Simplify by multiplying through.

Add and .

Apply the distributive property.

Multiply.

Multiply by .

Multiply by .

Add and .

Subtract from .

Subtract from .

Simplify the determinant.

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

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

Multiply by by adding the exponents.

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

Multiply by .

Rewrite using the commutative property of multiplication.

Multiply by by adding the exponents.

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Simplify by adding terms.

Add and .

Subtract from .

Add and .

The determinant of is .

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.

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 .

Add and .

Multiply by .

Combine the opposite terms in .

Add and .

Add and .

The determinant of is .

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

Simplify the determinant.

Simplify each term.

Apply the distributive property.

Multiply by .

Multiply by .

Multiply by .

Simplify by adding terms.

Combine the opposite terms in .

Add and .

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 .

Apply the distributive property.

Multiply .

Multiply by .

Multiply by .

Multiply by .

Apply the distributive property.

Move to the left of .

Multiply by .

Simplify by adding terms.

Combine the opposite terms in .

Subtract from .

Add and .

Multiply by .

Subtract from .

Add and .

Simplify the determinant.

Apply the distributive property.

Multiply by .

The determinant of is .

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.

Apply the distributive property.

Multiply by .

Multiply by .

Multiply by .

Subtract from .

The determinant of is .

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

Simplify the determinant.

Apply the distributive property.

Multiply .

Multiply by .

Multiply by .

Multiply by .

Simplify each term.

Apply the distributive property.

Multiply by .

Multiply by .

Multiply by .

Simplify terms.

Combine the opposite terms in .

Add and .

Add and .

Apply the distributive property.

Simplify the expression.

Rewrite using the commutative property of multiplication.

Multiply by .

Multiply by by adding the exponents.

Move .

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 the expression.

Add and .

Multiply by .

Subtract from .

Add and .

Simplify the determinant.

Apply the distributive property.

Simplify.

Multiply by .

Multiply by .

Multiply by .

The determinant of is .

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.

Multiply .

Multiply by .

Multiply by .

Multiply by .

Apply the distributive property.

Move to the left of .

Multiply by .

Add and .

The determinant of is .

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

Simplify the determinant.

Apply the distributive property.

Multiply .

Multiply by .

Multiply by .

Multiply by .

Simplify each term.

Multiply by .

Apply the distributive property.

Multiply .

Multiply by .

Multiply by .

Multiply by .

Apply the distributive property.

Move to the left of .

Multiply by .

Simplify terms.

Combine the opposite terms in .

Subtract from .

Add and .

Apply the distributive property.

Simplify the expression.

Rewrite using the commutative property of multiplication.

Multiply by .

Multiply by by adding the exponents.

Move .

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 the expression.

Add and .

Multiply by .

Add and .

Subtract from .

Simplify the determinant.

Apply the distributive property.

Simplify.

Multiply by .

Multiply by .

Multiply by .

Combine the opposite terms in .

Add and .

Add and .

Add and .

Add and .

Subtract from .

Subtract from .

Simplify by adding numbers.

Add and .

Add and .