# Linear Algebra Examples

Step 1

Set up the formula to find the characteristic equation .

Step 2

The identity matrix or unit matrix of size is the square matrix with ones on the main diagonal and zeros elsewhere.

Step 3

Step 3.1

Substitute for .

Step 3.2

Substitute for .

Step 4

Step 4.1

Simplify each term.

Step 4.1.1

Multiply by each element of the matrix.

Step 4.1.2

Simplify each element in the matrix.

Step 4.1.2.1

Multiply by .

Step 4.1.2.2

Multiply .

Step 4.1.2.2.1

Multiply by .

Step 4.1.2.2.2

Multiply by .

Step 4.1.2.3

Multiply .

Step 4.1.2.3.1

Multiply by .

Step 4.1.2.3.2

Multiply by .

Step 4.1.2.4

Multiply .

Step 4.1.2.4.1

Multiply by .

Step 4.1.2.4.2

Multiply by .

Step 4.1.2.5

Multiply by .

Step 4.1.2.6

Multiply .

Step 4.1.2.6.1

Multiply by .

Step 4.1.2.6.2

Multiply by .

Step 4.1.2.7

Multiply .

Step 4.1.2.7.1

Multiply by .

Step 4.1.2.7.2

Multiply by .

Step 4.1.2.8

Multiply .

Step 4.1.2.8.1

Multiply by .

Step 4.1.2.8.2

Multiply by .

Step 4.1.2.9

Multiply by .

Step 4.2

Add the corresponding elements.

Step 4.3

Simplify each element.

Step 4.3.1

Add and .

Step 4.3.2

Add and .

Step 4.3.3

Add and .

Step 4.3.4

Add and .

Step 4.3.5

Add and .

Step 4.3.6

Add and .

Step 5

Step 5.1

Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in column by its cofactor and add.

Step 5.1.1

Consider the corresponding sign chart.

Step 5.1.2

The cofactor is the minor with the sign changed if the indices match a position on the sign chart.

Step 5.1.3

The minor for is the determinant with row and column deleted.

Step 5.1.4

Multiply element by its cofactor.

Step 5.1.5

The minor for is the determinant with row and column deleted.

Step 5.1.6

Multiply element by its cofactor.

Step 5.1.7

The minor for is the determinant with row and column deleted.

Step 5.1.8

Multiply element by its cofactor.

Step 5.1.9

Add the terms together.

Step 5.2

Multiply by .

Step 5.3

Evaluate .

Step 5.3.1

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

Step 5.3.2

Simplify the determinant.

Step 5.3.2.1

Simplify each term.

Step 5.3.2.1.1

Multiply by .

Step 5.3.2.1.2

Multiply .

Step 5.3.2.1.2.1

Multiply by .

Step 5.3.2.1.2.2

Multiply by .

Step 5.3.2.2

Add and .

Step 5.4

Evaluate .

Step 5.4.1

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

Step 5.4.2

Simplify the determinant.

Step 5.4.2.1

Simplify each term.

Step 5.4.2.1.1

Expand using the FOIL Method.

Step 5.4.2.1.1.1

Apply the distributive property.

Step 5.4.2.1.1.2

Apply the distributive property.

Step 5.4.2.1.1.3

Apply the distributive property.

Step 5.4.2.1.2

Simplify and combine like terms.

Step 5.4.2.1.2.1

Simplify each term.

Step 5.4.2.1.2.1.1

Multiply by .

Step 5.4.2.1.2.1.2

Multiply .

Step 5.4.2.1.2.1.2.1

Multiply by .

Step 5.4.2.1.2.1.2.2

Multiply by .

Step 5.4.2.1.2.1.3

Multiply .

Step 5.4.2.1.2.1.3.1

Multiply by .

Step 5.4.2.1.2.1.3.2

Multiply by .

Step 5.4.2.1.2.1.4

Rewrite using the commutative property of multiplication.

Step 5.4.2.1.2.1.5

Multiply by by adding the exponents.

Step 5.4.2.1.2.1.5.1

Move .

Step 5.4.2.1.2.1.5.2

Multiply by .

Step 5.4.2.1.2.1.6

Multiply by .

Step 5.4.2.1.2.1.7

Multiply by .

Step 5.4.2.1.2.2

Add and .

Step 5.4.2.1.3

Multiply .

Step 5.4.2.1.3.1

Multiply by .

Step 5.4.2.1.3.2

Multiply by .

Step 5.4.2.2

Add and .

Step 5.4.2.3

Reorder and .

Step 5.5

Simplify the determinant.

Step 5.5.1

Add and .

Step 5.5.2

Simplify each term.

Step 5.5.2.1

Apply the distributive property.

Step 5.5.2.2

Multiply by .

Step 5.5.2.3

Multiply by .

Step 5.5.2.4

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

Step 5.5.2.5

Simplify each term.

Step 5.5.2.5.1

Multiply by .

Step 5.5.2.5.2

Multiply by .

Step 5.5.2.5.3

Multiply by .

Step 5.5.2.5.4

Multiply by by adding the exponents.

Step 5.5.2.5.4.1

Move .

Step 5.5.2.5.4.2

Multiply by .

Step 5.5.2.5.4.2.1

Raise to the power of .

Step 5.5.2.5.4.2.2

Use the power rule to combine exponents.

Step 5.5.2.5.4.3

Add and .

Step 5.5.2.5.5

Rewrite using the commutative property of multiplication.

Step 5.5.2.5.6

Multiply by by adding the exponents.

Step 5.5.2.5.6.1

Move .

Step 5.5.2.5.6.2

Multiply by .

Step 5.5.2.5.7

Multiply by .

Step 5.5.2.5.8

Multiply by .

Step 5.5.2.6

Subtract from .

Step 5.5.2.7

Subtract from .

Step 5.5.3

Combine the opposite terms in .

Step 5.5.3.1

Add and .

Step 5.5.3.2

Add and .

Step 5.5.4

Subtract from .

Step 5.5.5

Move .

Step 5.5.6

Reorder and .

Step 6

Set the characteristic polynomial equal to to find the eigenvalues .

Step 7

Step 7.1

Factor the left side of the equation.

Step 7.1.1

Factor out of .

Step 7.1.1.1

Factor out of .

Step 7.1.1.2

Factor out of .

Step 7.1.1.3

Factor out of .

Step 7.1.1.4

Factor out of .

Step 7.1.1.5

Factor out of .

Step 7.1.2

Factor.

Step 7.1.2.1

Factor using the AC method.

Step 7.1.2.1.1

Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .

Step 7.1.2.1.2

Write the factored form using these integers.

Step 7.1.2.2

Remove unnecessary parentheses.

Step 7.2

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

Step 7.3

Set equal to .

Step 7.4

Set equal to and solve for .

Step 7.4.1

Set equal to .

Step 7.4.2

Add to both sides of the equation.

Step 7.5

Set equal to and solve for .

Step 7.5.1

Set equal to .

Step 7.5.2

Subtract from both sides of the equation.

Step 7.6

The final solution is all the values that make true.