# Linear Algebra Examples

Step 1
Assign the matrix the name to simplify the descriptions throughout the problem.
Step 2
Set up the formula to find the characteristic equation .
Step 3
Substitute the known values in the formula.
Step 4
Subtract the eigenvalue times the identity matrix from the original matrix.
Multiply by each element of the matrix.
Simplify each element in the matrix.
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Simplify each element of the matrix .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Step 5
Find the determinant of .
Set up the determinant by breaking it into smaller components.
Since the matrix is multiplied by , the determinant is .
Since the matrix is multiplied by , the determinant is .
Find the determinant of .
The determinant of a matrix can be found using the formula .
Simplify the determinant.
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 .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Move .
Multiply by .
Multiply by .
Multiply by .
Subtract from .
Multiply by .
Simplify by multiplying through.
Subtract from .
Apply the distributive property.
Simplify.
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Move .
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Multiply by .
Simplify each term.
Multiply by by adding the exponents.
Move .
Multiply by .
Multiply by .
Combine the opposite terms in .
Factor the characteristic polynomial.
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Let . Substitute for all occurrences of .
Factor by grouping.
Reorder terms.
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Factor out of .
Rewrite as plus
Apply the distributive property.
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
Factor out the greatest common factor (GCF) from each group.
Factor the polynomial by factoring out the greatest common factor, .
Factor.
Replace all occurrences of with .
Remove unnecessary parentheses.
Step 6
Set the characteristic polynomial equal to to find the eigenvalues .
Step 7
Find the roots of by solving for .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set equal to .
Set equal to and solve for .
Set equal to .
Solve for .
Add to both sides of the equation.
Divide each term in by and simplify.
Divide each term in by .
Simplify the left side.
Dividing two negative values results in a positive value.
Divide by .
Simplify the right side.
Divide by .
Set equal to and solve for .
Set equal to .
Add to both sides of the equation.
The final solution is all the values that make true.
Step 8
The eigenvector for is equal to the null space of the matrix minus the eigenvalue times the identity matrix.
Step 9
Substitute the known values into the formula.
Step 10
Simplify the matrix expression.
Multiply by each element of the matrix.
Simplify each element in the matrix.
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Simplify each element of the matrix .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Step 11
Find the reduced row echelon form of the matrix.
Perform the row operation on (row ) in order to convert some elements in the row to .
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Step 12
Use the result matrix to declare the final solutions to the system of equations.
Step 13
This expression is the solution set for the system of equations.
Step 14
Decompose a solution vector by re-arranging each equation represented in the row-reduced form of the augmented matrix by solving for the dependent variable in each row yields the vector equality.
Step 15
Express the vector as a linear combination of column vector using the properties of vector column addition.
Step 16
The null space of the set is the set of vectors created from the free variables of the system.
Step 17
The eigenvector for is equal to the null space of the matrix minus the eigenvalue times the identity matrix.
Step 18
Substitute the known values into the formula.
Step 19
Simplify the matrix expression.
Multiply by each element of the matrix.
Simplify each element in the matrix.
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Simplify each element of the matrix .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Step 20
Find the reduced row echelon form of the matrix.
Perform the row operation on (row ) in order to convert some elements in the row to .
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Step 21
Exchange row and row to organize the zeros into position.
Step 22
Use the result matrix to declare the final solutions to the system of equations.
Step 23
This expression is the solution set for the system of equations.
Step 24
Decompose a solution vector by re-arranging each equation represented in the row-reduced form of the augmented matrix by solving for the dependent variable in each row yields the vector equality.
Step 25
Express the vector as a linear combination of column vector using the properties of vector column addition.
Step 26
The null space of the set is the set of vectors created from the free variables of the system.
Step 27
The eigenvector for is equal to the null space of the matrix minus the eigenvalue times the identity matrix.
Step 28
Substitute the known values into the formula.
Step 29
Simplify the matrix expression.
Multiply by each element of the matrix.
Simplify each element in the matrix.
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Simplify each element of the matrix .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Step 30
Find the reduced row echelon form of the matrix.
Perform the row operation on (row ) in order to convert some elements in the row to .
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Exchange row and row to organize the zeros into position.
Perform the row operation on (row ) in order to convert some elements in the row to .
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Step 31
Use the result matrix to declare the final solutions to the system of equations.
Step 32
This expression is the solution set for the system of equations.
Step 33
Decompose a solution vector by re-arranging each equation represented in the row-reduced form of the augmented matrix by solving for the dependent variable in each row yields the vector equality.
Step 34
Express the vector as a linear combination of column vector using the properties of vector column addition.
Step 35
The null space of the set is the set of vectors created from the free variables of the system.
Step 36
The eigenspace of is the union of the vector space for each eigenvalue.
Step 37