# Find the Characteristic Equation [[minus two , six , minus seven],[three , minus nine , ten],[minus one , three , minus three]]

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 .

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 .

Subtract from .

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

Simplify 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 .

Multiply by .

Simplify by adding terms.

Subtract from .

Subtract from .

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.

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 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.

Subtract from .

Apply the distributive property.

Multiply.

Multiply by .

Multiply by .

Add and .

Add and .

Subtract from .

Combine the opposite terms in .

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 .

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