# Algebra Examples

Step 1
Find the eigenvalues.
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 .
Step 1.4.1.2.4.1
Multiply by .
Step 1.4.1.2.4.2
Multiply by .
Step 1.4.1.2.5
Multiply by .
Step 1.4.1.2.6
Multiply .
Step 1.4.1.2.6.1
Multiply by .
Step 1.4.1.2.6.2
Multiply by .
Step 1.4.1.2.7
Multiply .
Step 1.4.1.2.7.1
Multiply by .
Step 1.4.1.2.7.2
Multiply by .
Step 1.4.1.2.8
Multiply .
Step 1.4.1.2.8.1
Multiply by .
Step 1.4.1.2.8.2
Multiply by .
Step 1.4.1.2.9
Multiply by .
Step 1.4.2
Step 1.4.3
Simplify each element.
Step 1.4.3.1
Step 1.4.3.2
Step 1.4.3.3
Step 1.4.3.4
Step 1.4.3.5
Step 1.4.3.6
Step 1.4.3.7
Subtract from .
Step 1.5
Find the determinant.
Step 1.5.1
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 1.5.1.1
Consider the corresponding sign chart.
Step 1.5.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 1.5.1.3
The minor for is the determinant with row and column deleted.
Step 1.5.1.4
Multiply element by its cofactor.
Step 1.5.1.5
The minor for is the determinant with row and column deleted.
Step 1.5.1.6
Multiply element by its cofactor.
Step 1.5.1.7
The minor for is the determinant with row and column deleted.
Step 1.5.1.8
Multiply element by its cofactor.
Step 1.5.1.9
Step 1.5.2
Multiply by .
Step 1.5.3
Multiply by .
Step 1.5.4
Evaluate .
Step 1.5.4.1
The determinant of a matrix can be found using the formula .
Step 1.5.4.2
Simplify the determinant.
Step 1.5.4.2.1
Simplify each term.
Step 1.5.4.2.1.1
Expand using the FOIL Method.
Step 1.5.4.2.1.1.1
Apply the distributive property.
Step 1.5.4.2.1.1.2
Apply the distributive property.
Step 1.5.4.2.1.1.3
Apply the distributive property.
Step 1.5.4.2.1.2
Simplify and combine like terms.
Step 1.5.4.2.1.2.1
Simplify each term.
Step 1.5.4.2.1.2.1.1
Multiply by .
Step 1.5.4.2.1.2.1.2
Multiply by .
Step 1.5.4.2.1.2.1.3
Multiply by .
Step 1.5.4.2.1.2.1.4
Rewrite using the commutative property of multiplication.
Step 1.5.4.2.1.2.1.5
Multiply by by adding the exponents.
Step 1.5.4.2.1.2.1.5.1
Move .
Step 1.5.4.2.1.2.1.5.2
Multiply by .
Step 1.5.4.2.1.2.1.6
Multiply by .
Step 1.5.4.2.1.2.1.7
Multiply by .
Step 1.5.4.2.1.2.2
Subtract from .
Step 1.5.4.2.1.3
Multiply by .
Step 1.5.4.2.2
Subtract from .
Step 1.5.4.2.3
Reorder and .
Step 1.5.5
Simplify the determinant.
Step 1.5.5.1
Combine the opposite terms in .
Step 1.5.5.1.1
Step 1.5.5.1.2
Subtract from .
Step 1.5.5.2
Apply the distributive property.
Step 1.5.5.3
Simplify.
Step 1.5.5.3.1
Multiply by by adding the exponents.
Step 1.5.5.3.1.1
Move .
Step 1.5.5.3.1.2
Multiply by .
Step 1.5.5.3.1.2.1
Raise to the power of .
Step 1.5.5.3.1.2.2
Use the power rule to combine exponents.
Step 1.5.5.3.1.3
Step 1.5.5.3.2
Rewrite using the commutative property of multiplication.
Step 1.5.5.3.3
Multiply by .
Step 1.5.5.4
Simplify each term.
Step 1.5.5.4.1
Multiply by by adding the exponents.
Step 1.5.5.4.1.1
Move .
Step 1.5.5.4.1.2
Multiply by .
Step 1.5.5.4.2
Multiply by .
Step 1.6
Set the characteristic polynomial equal to to find the eigenvalues .
Step 1.7
Solve for .
Step 1.7.1
Factor the left side of the equation.
Step 1.7.1.1
Factor out of .
Step 1.7.1.1.1
Factor out of .
Step 1.7.1.1.2
Factor out of .
Step 1.7.1.1.3
Factor out of .
Step 1.7.1.1.4
Factor out of .
Step 1.7.1.1.5
Factor out of .
Step 1.7.1.2
Factor.
Step 1.7.1.2.1
Factor using the AC method.
Step 1.7.1.2.1.1
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 1.7.1.2.1.2
Write the factored form using these integers.
Step 1.7.1.2.2
Remove unnecessary parentheses.
Step 1.7.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 1.7.3
Set equal to .
Step 1.7.4
Set equal to and solve for .
Step 1.7.4.1
Set equal to .
Step 1.7.4.2
Add to both sides of the equation.
Step 1.7.5
Set equal to and solve for .
Step 1.7.5.1
Set equal to .
Step 1.7.5.2
Subtract from both sides of the equation.
Step 1.7.6
The final solution is all the values that make true.
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
Find the eigenvector using the eigenvalue .
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 by .
Step 3.2.1.2.3
Multiply by .
Step 3.2.1.2.4
Multiply by .
Step 3.2.1.2.5
Multiply by .
Step 3.2.1.2.6
Multiply by .
Step 3.2.1.2.7
Multiply by .
Step 3.2.1.2.8
Multiply by .
Step 3.2.1.2.9
Multiply by .
Step 3.2.2
Adding any matrix to a null matrix is the matrix itself.
Step 3.2.2.1
Step 3.2.2.2
Simplify each element.
Step 3.2.2.2.1
Step 3.2.2.2.2
Step 3.2.2.2.3
Step 3.2.2.2.4
Step 3.2.2.2.5
Step 3.2.2.2.6
Step 3.2.2.2.7
Step 3.2.2.2.8
Step 3.2.2.2.9
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 3.3.2.3
Perform the row operation to make the entry at a .
Step 3.3.2.3.1
Perform the row operation to make the entry at a .
Step 3.3.2.3.2
Simplify .
Step 3.3.2.4
Multiply each element of by to make the entry at a .
Step 3.3.2.4.1
Multiply each element of by to make the entry at a .
Step 3.3.2.4.2
Simplify .
Step 3.3.2.5
Perform the row operation to make the entry at a .
Step 3.3.2.5.1
Perform the row operation to make the entry at a .
Step 3.3.2.5.2
Simplify .
Step 3.3.2.6
Perform the row operation to make the entry at a .
Step 3.3.2.6.1
Perform the row operation to make the entry at a .
Step 3.3.2.6.2
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
Find the eigenvector using the eigenvalue .
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 by .
Step 4.2.1.2.3
Multiply by .
Step 4.2.1.2.4
Multiply by .
Step 4.2.1.2.5
Multiply by .
Step 4.2.1.2.6
Multiply by .
Step 4.2.1.2.7
Multiply by .
Step 4.2.1.2.8
Multiply by .
Step 4.2.1.2.9
Multiply by .
Step 4.2.2
Step 4.2.3
Simplify each element.
Step 4.2.3.1
Subtract from .
Step 4.2.3.2
Step 4.2.3.3
Step 4.2.3.4
Step 4.2.3.5
Subtract from .
Step 4.2.3.6
Step 4.2.3.7
Step 4.2.3.8
Step 4.2.3.9
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 4.3.2.3
Perform the row operation to make the entry at a .
Step 4.3.2.3.1
Perform the row operation to make the entry at a .
Step 4.3.2.3.2
Simplify .
Step 4.3.2.4
Swap with to put a nonzero entry at .
Step 4.3.2.5
Multiply each element of by to make the entry at a .
Step 4.3.2.5.1
Multiply each element of by to make the entry at a .
Step 4.3.2.5.2
Simplify .
Step 4.3.2.6
Perform the row operation to make the entry at a .
Step 4.3.2.6.1
Perform the row operation to make the entry at a .
Step 4.3.2.6.2
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
Find the eigenvector using the eigenvalue .
Step 5.1
Substitute the known values into the formula.
Step 5.2
Simplify.
Step 5.2.1
Simplify each term.
Step 5.2.1.1
Multiply by each element of the matrix.
Step 5.2.1.2
Simplify each element in the matrix.
Step 5.2.1.2.1
Multiply by .
Step 5.2.1.2.2
Multiply by .
Step 5.2.1.2.3
Multiply by .
Step 5.2.1.2.4
Multiply by .
Step 5.2.1.2.5
Multiply by .
Step 5.2.1.2.6
Multiply by .
Step 5.2.1.2.7
Multiply by .
Step 5.2.1.2.8
Multiply by .
Step 5.2.1.2.9
Multiply by .
Step 5.2.2
Step 5.2.3
Simplify each element.
Step 5.2.3.1
Step 5.2.3.2
Step 5.2.3.3
Step 5.2.3.4
Step 5.2.3.5
Step 5.2.3.6
Step 5.2.3.7
Step 5.2.3.8
Step 5.2.3.9
Step 5.3
Find the null space when .
Step 5.3.1
Write as an augmented matrix for .
Step 5.3.2
Find the reduced row echelon form.
Step 5.3.2.1
Multiply each element of by to make the entry at a .
Step 5.3.2.1.1
Multiply each element of by to make the entry at a .
Step 5.3.2.1.2
Simplify .
Step 5.3.2.2
Perform the row operation to make the entry at a .
Step 5.3.2.2.1
Perform the row operation to make the entry at a .
Step 5.3.2.2.2
Simplify .
Step 5.3.2.3
Perform the row operation to make the entry at a .
Step 5.3.2.3.1
Perform the row operation to make the entry at a .
Step 5.3.2.3.2
Simplify .
Step 5.3.2.4
Swap with to put a nonzero entry at .
Step 5.3.2.5
Multiply each element of by to make the entry at a .
Step 5.3.2.5.1
Multiply each element of by to make the entry at a .
Step 5.3.2.5.2
Simplify .
Step 5.3.3
Use the result matrix to declare the final solution to the system of equations.
Step 5.3.4
Write a solution vector by solving in terms of the free variables in each row.
Step 5.3.5
Write the solution as a linear combination of vectors.
Step 5.3.6
Write as a solution set.
Step 5.3.7
The solution is the set of vectors created from the free variables of the system.
Step 6
The eigenspace of is the list of the vector space for each eigenvalue.