# 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 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 5

Step 5.1

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

Step 5.2

Simplify the determinant.

Step 5.2.1

Simplify each term.

Step 5.2.1.1

Expand using the FOIL Method.

Step 5.2.1.1.1

Apply the distributive property.

Step 5.2.1.1.2

Apply the distributive property.

Step 5.2.1.1.3

Apply the distributive property.

Step 5.2.1.2

Simplify and combine like terms.

Step 5.2.1.2.1

Simplify each term.

Step 5.2.1.2.1.1

Multiply by .

Step 5.2.1.2.1.2

Multiply by .

Step 5.2.1.2.1.3

Multiply by .

Step 5.2.1.2.1.4

Rewrite using the commutative property of multiplication.

Step 5.2.1.2.1.5

Multiply by by adding the exponents.

Step 5.2.1.2.1.5.1

Move .

Step 5.2.1.2.1.5.2

Multiply by .

Step 5.2.1.2.1.6

Multiply by .

Step 5.2.1.2.1.7

Multiply by .

Step 5.2.1.2.2

Subtract from .

Step 5.2.1.3

Multiply by .

Step 5.2.2

Subtract from .

Step 5.2.3

Reorder and .

Step 6

Set the characteristic polynomial equal to to find the eigenvalues .

Step 7

Step 7.1

Use the quadratic formula to find the solutions.

Step 7.2

Substitute the values , , and into the quadratic formula and solve for .

Step 7.3

Simplify.

Step 7.3.1

Simplify the numerator.

Step 7.3.1.1

Raise to the power of .

Step 7.3.1.2

Multiply .

Step 7.3.1.2.1

Multiply by .

Step 7.3.1.2.2

Multiply by .

Step 7.3.1.3

Subtract from .

Step 7.3.2

Multiply by .

Step 7.4

The final answer is the combination of both solutions.

Step 8

The result can be shown in multiple forms.

Exact Form:

Decimal Form: