Linear Algebra Examples

Find the Eigenvalues [[1,0,1,1],[0,1,1,1],[1,1,1,0],[1,1,0,1]]
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
Substitute the known values into .
Tap for more steps...
Step 3.1
Substitute for .
Step 3.2
Substitute for .
Step 4
Simplify.
Tap for more steps...
Step 4.1
Simplify each term.
Tap for more steps...
Step 4.1.1
Multiply by each element of the matrix.
Step 4.1.2
Simplify each element in the matrix.
Tap for more steps...
Step 4.1.2.1
Multiply by .
Step 4.1.2.2
Multiply .
Tap for more steps...
Step 4.1.2.2.1
Multiply by .
Step 4.1.2.2.2
Multiply by .
Step 4.1.2.3
Multiply .
Tap for more steps...
Step 4.1.2.3.1
Multiply by .
Step 4.1.2.3.2
Multiply by .
Step 4.1.2.4
Multiply .
Tap for more steps...
Step 4.1.2.4.1
Multiply by .
Step 4.1.2.4.2
Multiply by .
Step 4.1.2.5
Multiply .
Tap for more steps...
Step 4.1.2.5.1
Multiply by .
Step 4.1.2.5.2
Multiply by .
Step 4.1.2.6
Multiply by .
Step 4.1.2.7
Multiply .
Tap for more steps...
Step 4.1.2.7.1
Multiply by .
Step 4.1.2.7.2
Multiply by .
Step 4.1.2.8
Multiply .
Tap for more steps...
Step 4.1.2.8.1
Multiply by .
Step 4.1.2.8.2
Multiply by .
Step 4.1.2.9
Multiply .
Tap for more steps...
Step 4.1.2.9.1
Multiply by .
Step 4.1.2.9.2
Multiply by .
Step 4.1.2.10
Multiply .
Tap for more steps...
Step 4.1.2.10.1
Multiply by .
Step 4.1.2.10.2
Multiply by .
Step 4.1.2.11
Multiply by .
Step 4.1.2.12
Multiply .
Tap for more steps...
Step 4.1.2.12.1
Multiply by .
Step 4.1.2.12.2
Multiply by .
Step 4.1.2.13
Multiply .
Tap for more steps...
Step 4.1.2.13.1
Multiply by .
Step 4.1.2.13.2
Multiply by .
Step 4.1.2.14
Multiply .
Tap for more steps...
Step 4.1.2.14.1
Multiply by .
Step 4.1.2.14.2
Multiply by .
Step 4.1.2.15
Multiply .
Tap for more steps...
Step 4.1.2.15.1
Multiply by .
Step 4.1.2.15.2
Multiply by .
Step 4.1.2.16
Multiply by .
Step 4.2
Add the corresponding elements.
Step 4.3
Simplify each element.
Tap for more steps...
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 4.3.7
Add and .
Step 4.3.8
Add and .
Step 4.3.9
Add and .
Step 4.3.10
Add and .
Step 4.3.11
Add and .
Step 4.3.12
Add and .
Step 5
Find the determinant.
Tap for more steps...
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 row by its cofactor and add.
Tap for more steps...
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
The minor for is the determinant with row and column deleted.
Step 5.1.10
Multiply element by its cofactor.
Step 5.1.11
Add the terms together.
Step 5.2
Multiply by .
Step 5.3
Evaluate .
Tap for more steps...
Step 5.3.1
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.
Tap for more steps...
Step 5.3.1.1
Consider the corresponding sign chart.
Step 5.3.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 5.3.1.3
The minor for is the determinant with row and column deleted.
Step 5.3.1.4
Multiply element by its cofactor.
Step 5.3.1.5
The minor for is the determinant with row and column deleted.
Step 5.3.1.6
Multiply element by its cofactor.
Step 5.3.1.7
The minor for is the determinant with row and column deleted.
Step 5.3.1.8
Multiply element by its cofactor.
Step 5.3.1.9
Add the terms together.
Step 5.3.2
Multiply by .
Step 5.3.3
Evaluate .
Tap for more steps...
Step 5.3.3.1
The determinant of a matrix can be found using the formula .
Step 5.3.3.2
Simplify the determinant.
Tap for more steps...
Step 5.3.3.2.1
Simplify each term.
Tap for more steps...
Step 5.3.3.2.1.1
Multiply by .
Step 5.3.3.2.1.2
Multiply by .
Step 5.3.3.2.2
Add and .
Step 5.3.3.2.3
Reorder and .
Step 5.3.4
Evaluate .
Tap for more steps...
Step 5.3.4.1
The determinant of a matrix can be found using the formula .
Step 5.3.4.2
Simplify the determinant.
Tap for more steps...
Step 5.3.4.2.1
Simplify each term.
Tap for more steps...
Step 5.3.4.2.1.1
Expand using the FOIL Method.
Tap for more steps...
Step 5.3.4.2.1.1.1
Apply the distributive property.
Step 5.3.4.2.1.1.2
Apply the distributive property.
Step 5.3.4.2.1.1.3
Apply the distributive property.
Step 5.3.4.2.1.2
Simplify and combine like terms.
Tap for more steps...
Step 5.3.4.2.1.2.1
Simplify each term.
Tap for more steps...
Step 5.3.4.2.1.2.1.1
Multiply by .
Step 5.3.4.2.1.2.1.2
Multiply by .
Step 5.3.4.2.1.2.1.3
Multiply by .
Step 5.3.4.2.1.2.1.4
Rewrite using the commutative property of multiplication.
Step 5.3.4.2.1.2.1.5
Multiply by by adding the exponents.
Tap for more steps...
Step 5.3.4.2.1.2.1.5.1
Move .
Step 5.3.4.2.1.2.1.5.2
Multiply by .
Step 5.3.4.2.1.2.1.6
Multiply by .
Step 5.3.4.2.1.2.1.7
Multiply by .
Step 5.3.4.2.1.2.2
Subtract from .
Step 5.3.4.2.1.3
Multiply by .
Step 5.3.4.2.2
Combine the opposite terms in .
Tap for more steps...
Step 5.3.4.2.2.1
Subtract from .
Step 5.3.4.2.2.2
Add and .
Step 5.3.4.2.3
Reorder and .
Step 5.3.5
Simplify the determinant.
Tap for more steps...
Step 5.3.5.1
Add and .
Step 5.3.5.2
Simplify each term.
Tap for more steps...
Step 5.3.5.2.1
Apply the distributive property.
Step 5.3.5.2.2
Multiply .
Tap for more steps...
Step 5.3.5.2.2.1
Multiply by .
Step 5.3.5.2.2.2
Multiply by .
Step 5.3.5.2.3
Multiply by .
Step 5.3.5.2.4
Expand using the FOIL Method.
Tap for more steps...
Step 5.3.5.2.4.1
Apply the distributive property.
Step 5.3.5.2.4.2
Apply the distributive property.
Step 5.3.5.2.4.3
Apply the distributive property.
Step 5.3.5.2.5
Simplify and combine like terms.
Tap for more steps...
Step 5.3.5.2.5.1
Simplify each term.
Tap for more steps...
Step 5.3.5.2.5.1.1
Multiply by .
Step 5.3.5.2.5.1.2
Multiply by .
Step 5.3.5.2.5.1.3
Multiply by by adding the exponents.
Tap for more steps...
Step 5.3.5.2.5.1.3.1
Move .
Step 5.3.5.2.5.1.3.2
Multiply by .
Tap for more steps...
Step 5.3.5.2.5.1.3.2.1
Raise to the power of .
Step 5.3.5.2.5.1.3.2.2
Use the power rule to combine exponents.
Step 5.3.5.2.5.1.3.3
Add and .
Step 5.3.5.2.5.1.4
Rewrite using the commutative property of multiplication.
Step 5.3.5.2.5.1.5
Multiply by by adding the exponents.
Tap for more steps...
Step 5.3.5.2.5.1.5.1
Move .
Step 5.3.5.2.5.1.5.2
Multiply by .
Step 5.3.5.2.5.1.6
Multiply by .
Step 5.3.5.2.5.2
Add and .
Step 5.3.5.3
Subtract from .
Step 5.3.5.4
Move .
Step 5.3.5.5
Move .
Step 5.3.5.6
Reorder and .
Step 5.4
Evaluate .
Tap for more steps...
Step 5.4.1
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.
Tap for more steps...
Step 5.4.1.1
Consider the corresponding sign chart.
Step 5.4.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 5.4.1.3
The minor for is the determinant with row and column deleted.
Step 5.4.1.4
Multiply element by its cofactor.
Step 5.4.1.5
The minor for is the determinant with row and column deleted.
Step 5.4.1.6
Multiply element by its cofactor.
Step 5.4.1.7
The minor for is the determinant with row and column deleted.
Step 5.4.1.8
Multiply element by its cofactor.
Step 5.4.1.9
Add the terms together.
Step 5.4.2
Multiply by .
Step 5.4.3
Evaluate .
Tap for more steps...
Step 5.4.3.1
The determinant of a matrix can be found using the formula .
Step 5.4.3.2
Simplify the determinant.
Tap for more steps...
Step 5.4.3.2.1
Multiply by .
Step 5.4.3.2.2
Subtract from .
Step 5.4.3.2.3
Reorder and .
Step 5.4.4
Evaluate .
Tap for more steps...
Step 5.4.4.1
The determinant of a matrix can be found using the formula .
Step 5.4.4.2
Simplify the determinant.
Tap for more steps...
Step 5.4.4.2.1
Simplify each term.
Tap for more steps...
Step 5.4.4.2.1.1
Multiply by .
Step 5.4.4.2.1.2
Multiply by .
Step 5.4.4.2.2
Subtract from .
Step 5.4.5
Simplify the determinant.
Tap for more steps...
Step 5.4.5.1
Subtract from .
Step 5.4.5.2
Simplify each term.
Tap for more steps...
Step 5.4.5.2.1
Apply the distributive property.
Step 5.4.5.2.2
Multiply by .
Step 5.4.5.2.3
Multiply .
Tap for more steps...
Step 5.4.5.2.3.1
Multiply by .
Step 5.4.5.2.3.2
Multiply by .
Step 5.4.5.2.4
Expand using the FOIL Method.
Tap for more steps...
Step 5.4.5.2.4.1
Apply the distributive property.
Step 5.4.5.2.4.2
Apply the distributive property.
Step 5.4.5.2.4.3
Apply the distributive property.
Step 5.4.5.2.5
Simplify and combine like terms.
Tap for more steps...
Step 5.4.5.2.5.1
Simplify each term.
Tap for more steps...
Step 5.4.5.2.5.1.1
Multiply .
Tap for more steps...
Step 5.4.5.2.5.1.1.1
Multiply by .
Step 5.4.5.2.5.1.1.2
Multiply by .
Step 5.4.5.2.5.1.2
Multiply by .
Step 5.4.5.2.5.1.3
Rewrite using the commutative property of multiplication.
Step 5.4.5.2.5.1.4
Multiply by by adding the exponents.
Tap for more steps...
Step 5.4.5.2.5.1.4.1
Move .
Step 5.4.5.2.5.1.4.2
Multiply by .
Step 5.4.5.2.5.1.5
Multiply by .
Step 5.4.5.2.5.2
Add and .
Step 5.4.5.2.6
Multiply by .
Step 5.4.5.3
Add and .
Step 5.4.5.4
Move .
Step 5.4.5.5
Reorder and .
Step 5.5
Evaluate .
Tap for more steps...
Step 5.5.1
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.
Tap for more steps...
Step 5.5.1.1
Consider the corresponding sign chart.
Step 5.5.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 5.5.1.3
The minor for is the determinant with row and column deleted.
Step 5.5.1.4
Multiply element by its cofactor.
Step 5.5.1.5
The minor for is the determinant with row and column deleted.
Step 5.5.1.6
Multiply element by its cofactor.
Step 5.5.1.7
The minor for is the determinant with row and column deleted.
Step 5.5.1.8
Multiply element by its cofactor.
Step 5.5.1.9
Add the terms together.
Step 5.5.2
Multiply by .
Step 5.5.3
Evaluate .
Tap for more steps...
Step 5.5.3.1
The determinant of a matrix can be found using the formula .
Step 5.5.3.2
Simplify the determinant.
Tap for more steps...
Step 5.5.3.2.1
Simplify each term.
Tap for more steps...
Step 5.5.3.2.1.1
Multiply by .
Step 5.5.3.2.1.2
Apply the distributive property.
Step 5.5.3.2.1.3
Multiply by .
Step 5.5.3.2.1.4
Multiply .
Tap for more steps...
Step 5.5.3.2.1.4.1
Multiply by .
Step 5.5.3.2.1.4.2
Multiply by .
Step 5.5.3.2.2
Subtract from .
Step 5.5.3.2.3
Reorder and .
Step 5.5.4
Evaluate .
Tap for more steps...
Step 5.5.4.1
The determinant of a matrix can be found using the formula .
Step 5.5.4.2
Simplify the determinant.
Tap for more steps...
Step 5.5.4.2.1
Simplify each term.
Tap for more steps...
Step 5.5.4.2.1.1
Multiply by .
Step 5.5.4.2.1.2
Multiply by .
Step 5.5.4.2.2
Subtract from .
Step 5.5.5
Simplify the determinant.
Tap for more steps...
Step 5.5.5.1
Subtract from .
Step 5.5.5.2
Simplify each term.
Tap for more steps...
Step 5.5.5.2.1
Apply the distributive property.
Step 5.5.5.2.2
Multiply by .
Step 5.5.5.2.3
Multiply .
Tap for more steps...
Step 5.5.5.2.3.1
Multiply by .
Step 5.5.5.2.3.2
Multiply by .
Step 5.5.5.2.4
Expand using the FOIL Method.
Tap for more steps...
Step 5.5.5.2.4.1
Apply the distributive property.
Step 5.5.5.2.4.2
Apply the distributive property.
Step 5.5.5.2.4.3
Apply the distributive property.
Step 5.5.5.2.5
Simplify and combine like terms.
Tap for more steps...
Step 5.5.5.2.5.1
Simplify each term.
Tap for more steps...
Step 5.5.5.2.5.1.1
Rewrite as .
Step 5.5.5.2.5.1.2
Multiply by .
Step 5.5.5.2.5.1.3
Multiply by .
Step 5.5.5.2.5.1.4
Move to the left of .
Step 5.5.5.2.5.1.5
Rewrite as .
Step 5.5.5.2.5.2
Subtract from .
Step 5.5.5.2.6
Multiply by .
Step 5.5.5.3
Add and .
Step 5.5.5.4
Move .
Step 5.5.5.5
Reorder and .
Step 5.6
Simplify the determinant.
Tap for more steps...
Step 5.6.1
Add and .
Step 5.6.2
Simplify each term.
Tap for more steps...
Step 5.6.2.1
Expand by multiplying each term in the first expression by each term in the second expression.
Step 5.6.2.2
Simplify each term.
Tap for more steps...
Step 5.6.2.2.1
Multiply by .
Step 5.6.2.2.2
Multiply by .
Step 5.6.2.2.3
Multiply by .
Step 5.6.2.2.4
Multiply by .
Step 5.6.2.2.5
Rewrite using the commutative property of multiplication.
Step 5.6.2.2.6
Multiply by by adding the exponents.
Tap for more steps...
Step 5.6.2.2.6.1
Move .
Step 5.6.2.2.6.2
Multiply by .
Tap for more steps...
Step 5.6.2.2.6.2.1
Raise to the power of .
Step 5.6.2.2.6.2.2
Use the power rule to combine exponents.
Step 5.6.2.2.6.3
Add and .
Step 5.6.2.2.7
Multiply by .
Step 5.6.2.2.8
Multiply by .
Step 5.6.2.2.9
Rewrite using the commutative property of multiplication.
Step 5.6.2.2.10
Multiply by by adding the exponents.
Tap for more steps...
Step 5.6.2.2.10.1
Move .
Step 5.6.2.2.10.2
Multiply by .
Tap for more steps...
Step 5.6.2.2.10.2.1
Raise to the power of .
Step 5.6.2.2.10.2.2
Use the power rule to combine exponents.
Step 5.6.2.2.10.3
Add and .
Step 5.6.2.2.11
Multiply by .
Step 5.6.2.2.12
Rewrite using the commutative property of multiplication.
Step 5.6.2.2.13
Multiply by by adding the exponents.
Tap for more steps...
Step 5.6.2.2.13.1
Move .
Step 5.6.2.2.13.2
Multiply by .
Step 5.6.2.2.14
Multiply by .
Step 5.6.2.2.15
Multiply by .
Step 5.6.2.2.16
Multiply .
Tap for more steps...
Step 5.6.2.2.16.1
Multiply by .
Step 5.6.2.2.16.2
Multiply by .
Step 5.6.2.3
Combine the opposite terms in .
Tap for more steps...
Step 5.6.2.3.1
Add and .
Step 5.6.2.3.2
Add and .
Step 5.6.2.4
Subtract from .
Step 5.6.2.5
Add and .
Step 5.6.2.6
Multiply by .
Step 5.6.2.7
Apply the distributive property.
Step 5.6.2.8
Simplify.
Tap for more steps...
Step 5.6.2.8.1
Rewrite as .
Step 5.6.2.8.2
Multiply by .
Step 5.6.2.8.3
Multiply by .
Step 5.6.3
Subtract from .
Step 5.6.4
Subtract from .
Step 5.6.5
Subtract from .
Step 5.6.6
Add and .
Step 5.6.7
Subtract from .
Step 5.6.8
Move .
Step 5.6.9
Reorder and .
Step 6
Set the characteristic polynomial equal to to find the eigenvalues .
Step 7
Solve for .
Tap for more steps...
Step 7.1
Factor the left side of the equation.
Tap for more steps...
Step 7.1.1
Regroup terms.
Step 7.1.2
Factor out of .
Tap for more steps...
Step 7.1.2.1
Factor out of .
Step 7.1.2.2
Factor out of .
Step 7.1.2.3
Factor out of .
Step 7.1.3
Rewrite as .
Step 7.1.4
Factor.
Tap for more steps...
Step 7.1.4.1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 7.1.4.2
Remove unnecessary parentheses.
Step 7.1.5
Rewrite as .
Step 7.1.6
Let . Substitute for all occurrences of .
Step 7.1.7
Factor using the AC method.
Tap for more steps...
Step 7.1.7.1
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 7.1.7.2
Write the factored form using these integers.
Step 7.1.8
Replace all occurrences of with .
Step 7.1.9
Rewrite as .
Step 7.1.10
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 7.1.11
Factor out of .
Tap for more steps...
Step 7.1.11.1
Factor out of .
Step 7.1.11.2
Factor out of .
Step 7.1.12
Let . Substitute for all occurrences of .
Step 7.1.13
Factor using the AC method.
Tap for more steps...
Step 7.1.13.1
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 7.1.13.2
Write the factored form using these integers.
Step 7.1.14
Factor.
Tap for more steps...
Step 7.1.14.1
Replace all occurrences of with .
Step 7.1.14.2
Remove unnecessary parentheses.
Step 7.1.15
Combine exponents.
Tap for more steps...
Step 7.1.15.1
Raise to the power of .
Step 7.1.15.2
Raise to the power of .
Step 7.1.15.3
Use the power rule to combine exponents.
Step 7.1.15.4
Add and .
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 and solve for .
Tap for more steps...
Step 7.3.1
Set equal to .
Step 7.3.2
Subtract from both sides of the equation.
Step 7.4
Set equal to and solve for .
Tap for more steps...
Step 7.4.1
Set equal to .
Step 7.4.2
Solve for .
Tap for more steps...
Step 7.4.2.1
Set the equal to .
Step 7.4.2.2
Add to both sides of the equation.
Step 7.5
Set equal to and solve for .
Tap for more steps...
Step 7.5.1
Set equal to .
Step 7.5.2
Add to both sides of the equation.
Step 7.6
The final solution is all the values that make true.