# Linear Algebra Examples

Step 1

Step 1.1

Set up the formula to find the characteristic equation .

Step 1.2

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

Step 1.3

Substitute the known values into .

Step 1.3.1

Substitute for .

Step 1.3.2

Substitute for .

Step 1.4

Simplify.

Step 1.4.1

Simplify each term.

Step 1.4.1.1

Multiply by each element of the matrix.

Step 1.4.1.2

Simplify each element in the matrix.

Step 1.4.1.2.1

Multiply by .

Step 1.4.1.2.2

Multiply .

Step 1.4.1.2.2.1

Multiply by .

Step 1.4.1.2.2.2

Multiply by .

Step 1.4.1.2.3

Multiply .

Step 1.4.1.2.3.1

Multiply by .

Step 1.4.1.2.3.2

Multiply by .

Step 1.4.1.2.4

Multiply by .

Step 1.4.2

Add the corresponding elements.

Step 1.4.3

Simplify each element.

Step 1.4.3.1

Add and .

Step 1.4.3.2

Add and .

Step 1.5

Find the determinant.

Step 1.5.1

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

Step 1.5.2

Simplify the determinant.

Step 1.5.2.1

Simplify each term.

Step 1.5.2.1.1

Expand using the FOIL Method.

Step 1.5.2.1.1.1

Apply the distributive property.

Step 1.5.2.1.1.2

Apply the distributive property.

Step 1.5.2.1.1.3

Apply the distributive property.

Step 1.5.2.1.2

Simplify and combine like terms.

Step 1.5.2.1.2.1

Simplify each term.

Step 1.5.2.1.2.1.1

Multiply by .

Step 1.5.2.1.2.1.2

Multiply by .

Step 1.5.2.1.2.1.3

Multiply by .

Step 1.5.2.1.2.1.4

Rewrite using the commutative property of multiplication.

Step 1.5.2.1.2.1.5

Multiply by by adding the exponents.

Step 1.5.2.1.2.1.5.1

Move .

Step 1.5.2.1.2.1.5.2

Multiply by .

Step 1.5.2.1.2.1.6

Multiply by .

Step 1.5.2.1.2.1.7

Multiply by .

Step 1.5.2.1.2.2

Subtract from .

Step 1.5.2.1.3

Multiply by .

Step 1.5.2.2

Subtract from .

Step 1.5.2.3

Reorder and .

Step 1.6

Set the characteristic polynomial equal to to find the eigenvalues .

Step 1.7

Solve for .

Step 1.7.1

Use the quadratic formula to find the solutions.

Step 1.7.2

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

Step 1.7.3

Simplify.

Step 1.7.3.1

Simplify the numerator.

Step 1.7.3.1.1

Raise to the power of .

Step 1.7.3.1.2

Multiply .

Step 1.7.3.1.2.1

Multiply by .

Step 1.7.3.1.2.2

Multiply by .

Step 1.7.3.1.3

Add and .

Step 1.7.3.2

Multiply by .

Step 1.7.4

The final answer is the combination of both solutions.

Step 2

The eigenvector is equal to the null space of the matrix minus the eigenvalue times the identity matrix where is the null space and is the identity matrix.

Step 3

Step 3.1

Substitute the known values into the formula.

Step 3.2

Simplify.

Step 3.2.1

Simplify each term.

Step 3.2.1.1

Multiply by each element of the matrix.

Step 3.2.1.2

Simplify each element in the matrix.

Step 3.2.1.2.1

Multiply by .

Step 3.2.1.2.2

Multiply .

Step 3.2.1.2.2.1

Multiply by .

Step 3.2.1.2.2.2

Multiply by .

Step 3.2.1.2.3

Multiply .

Step 3.2.1.2.3.1

Multiply by .

Step 3.2.1.2.3.2

Multiply by .

Step 3.2.1.2.4

Multiply by .

Step 3.2.2

Add the corresponding elements.

Step 3.2.3

Simplify each element.

Step 3.2.3.1

Write as a fraction with a common denominator.

Step 3.2.3.2

Combine the numerators over the common denominator.

Step 3.2.3.3

Simplify the numerator.

Step 3.2.3.3.1

Apply the distributive property.

Step 3.2.3.3.2

Multiply by .

Step 3.2.3.3.3

Subtract from .

Step 3.2.3.4

Rewrite as .

Step 3.2.3.5

Factor out of .

Step 3.2.3.6

Factor out of .

Step 3.2.3.7

Move the negative in front of the fraction.

Step 3.2.3.8

Add and .

Step 3.2.3.9

Add and .

Step 3.2.3.10

To write as a fraction with a common denominator, multiply by .

Step 3.2.3.11

Combine and .

Step 3.2.3.12

Combine the numerators over the common denominator.

Step 3.2.3.13

Simplify the numerator.

Step 3.2.3.13.1

Multiply by .

Step 3.2.3.13.2

Apply the distributive property.

Step 3.2.3.13.3

Multiply by .

Step 3.2.3.13.4

Subtract from .

Step 3.3

Find the null space when .

Step 3.3.1

Write as an augmented matrix for .

Step 3.3.2

Find the reduced row echelon form.

Step 3.3.2.1

Multiply each element of by to make the entry at a .

Step 3.3.2.1.1

Multiply each element of by to make the entry at a .

Step 3.3.2.1.2

Simplify .

Step 3.3.2.2

Perform the row operation to make the entry at a .

Step 3.3.2.2.1

Perform the row operation to make the entry at a .

Step 3.3.2.2.2

Simplify .

Step 3.3.3

Use the result matrix to declare the final solution to the system of equations.

Step 3.3.4

Write a solution vector by solving in terms of the free variables in each row.

Step 3.3.5

Write the solution as a linear combination of vectors.

Step 3.3.6

Write as a solution set.

Step 3.3.7

The solution is the set of vectors created from the free variables of the system.

Step 4

Step 4.1

Substitute the known values into the formula.

Step 4.2

Simplify.

Step 4.2.1

Simplify each term.

Step 4.2.1.1

Multiply by each element of the matrix.

Step 4.2.1.2

Simplify each element in the matrix.

Step 4.2.1.2.1

Multiply by .

Step 4.2.1.2.2

Multiply .

Step 4.2.1.2.2.1

Multiply by .

Step 4.2.1.2.2.2

Multiply by .

Step 4.2.1.2.3

Multiply .

Step 4.2.1.2.3.1

Multiply by .

Step 4.2.1.2.3.2

Multiply by .

Step 4.2.1.2.4

Multiply by .

Step 4.2.2

Add the corresponding elements.

Step 4.2.3

Simplify each element.

Step 4.2.3.1

Write as a fraction with a common denominator.

Step 4.2.3.2

Combine the numerators over the common denominator.

Step 4.2.3.3

Simplify the numerator.

Step 4.2.3.3.1

Apply the distributive property.

Step 4.2.3.3.2

Multiply by .

Step 4.2.3.3.3

Multiply .

Step 4.2.3.3.3.1

Multiply by .

Step 4.2.3.3.3.2

Multiply by .

Step 4.2.3.3.4

Subtract from .

Step 4.2.3.4

Rewrite as .

Step 4.2.3.5

Factor out of .

Step 4.2.3.6

Factor out of .

Step 4.2.3.7

Move the negative in front of the fraction.

Step 4.2.3.8

Add and .

Step 4.2.3.9

Add and .

Step 4.2.3.10

To write as a fraction with a common denominator, multiply by .

Step 4.2.3.11

Combine and .

Step 4.2.3.12

Combine the numerators over the common denominator.

Step 4.2.3.13

Simplify the numerator.

Step 4.2.3.13.1

Multiply by .

Step 4.2.3.13.2

Apply the distributive property.

Step 4.2.3.13.3

Multiply by .

Step 4.2.3.13.4

Multiply .

Step 4.2.3.13.4.1

Multiply by .

Step 4.2.3.13.4.2

Multiply by .

Step 4.2.3.13.5

Subtract from .

Step 4.3

Find the null space when .

Step 4.3.1

Write as an augmented matrix for .

Step 4.3.2

Find the reduced row echelon form.

Step 4.3.2.1

Multiply each element of by to make the entry at a .

Step 4.3.2.1.1

Multiply each element of by to make the entry at a .

Step 4.3.2.1.2

Simplify .

Step 4.3.2.2

Perform the row operation to make the entry at a .

Step 4.3.2.2.1

Perform the row operation to make the entry at a .

Step 4.3.2.2.2

Simplify .

Step 4.3.3

Use the result matrix to declare the final solution to the system of equations.

Step 4.3.4

Write a solution vector by solving in terms of the free variables in each row.

Step 4.3.5

Write the solution as a linear combination of vectors.

Step 4.3.6

Write as a solution set.

Step 4.3.7

The solution is the set of vectors created from the free variables of the system.

Step 5

The eigenspace of is the list of the vector space for each eigenvalue.