Linear Algebra Examples

Find the Eigenvalues [[3,1,0,0],[1,3,0,0],[0,0,0,0],[0,0,0,0]]
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 .
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Step 3.1
Substitute for .
Step 3.2
Substitute for .
Step 4
Simplify.
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Step 4.1
Simplify each term.
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Step 4.1.1
Multiply by each element of the matrix.
Step 4.1.2
Simplify each element in the matrix.
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Step 4.1.2.1
Multiply by .
Step 4.1.2.2
Multiply .
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Step 4.1.2.2.1
Multiply by .
Step 4.1.2.2.2
Multiply by .
Step 4.1.2.3
Multiply .
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Step 4.1.2.3.1
Multiply by .
Step 4.1.2.3.2
Multiply by .
Step 4.1.2.4
Multiply .
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Step 4.1.2.4.1
Multiply by .
Step 4.1.2.4.2
Multiply by .
Step 4.1.2.5
Multiply .
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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 .
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Step 4.1.2.7.1
Multiply by .
Step 4.1.2.7.2
Multiply by .
Step 4.1.2.8
Multiply .
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Step 4.1.2.8.1
Multiply by .
Step 4.1.2.8.2
Multiply by .
Step 4.1.2.9
Multiply .
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Step 4.1.2.9.1
Multiply by .
Step 4.1.2.9.2
Multiply by .
Step 4.1.2.10
Multiply .
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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 .
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Step 4.1.2.12.1
Multiply by .
Step 4.1.2.12.2
Multiply by .
Step 4.1.2.13
Multiply .
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Step 4.1.2.13.1
Multiply by .
Step 4.1.2.13.2
Multiply by .
Step 4.1.2.14
Multiply .
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Step 4.1.2.14.1
Multiply by .
Step 4.1.2.14.2
Multiply by .
Step 4.1.2.15
Multiply .
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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.
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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
Subtract from .
Step 4.3.10
Add and .
Step 4.3.11
Add and .
Step 4.3.12
Add and .
Step 4.3.13
Add and .
Step 4.3.14
Subtract from .
Step 5
Find the determinant.
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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.
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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
Multiply by .
Step 5.4
Multiply by .
Step 5.5
Evaluate .
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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.
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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
Multiply by .
Step 5.5.4
Evaluate .
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Step 5.5.4.1
The determinant of a matrix can be found using the formula .
Step 5.5.4.2
Simplify the determinant.
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Step 5.5.4.2.1
Simplify each term.
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Step 5.5.4.2.1.1
Expand using the FOIL Method.
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Step 5.5.4.2.1.1.1
Apply the distributive property.
Step 5.5.4.2.1.1.2
Apply the distributive property.
Step 5.5.4.2.1.1.3
Apply the distributive property.
Step 5.5.4.2.1.2
Simplify and combine like terms.
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Step 5.5.4.2.1.2.1
Simplify each term.
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Step 5.5.4.2.1.2.1.1
Multiply by .
Step 5.5.4.2.1.2.1.2
Multiply by .
Step 5.5.4.2.1.2.1.3
Multiply by .
Step 5.5.4.2.1.2.1.4
Rewrite using the commutative property of multiplication.
Step 5.5.4.2.1.2.1.5
Multiply by by adding the exponents.
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Step 5.5.4.2.1.2.1.5.1
Move .
Step 5.5.4.2.1.2.1.5.2
Multiply by .
Step 5.5.4.2.1.2.1.6
Multiply by .
Step 5.5.4.2.1.2.1.7
Multiply by .
Step 5.5.4.2.1.2.2
Subtract from .
Step 5.5.4.2.1.3
Multiply by .
Step 5.5.4.2.2
Subtract from .
Step 5.5.4.2.3
Reorder and .
Step 5.5.5
Simplify the determinant.
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Step 5.5.5.1
Combine the opposite terms in .
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Step 5.5.5.1.1
Add and .
Step 5.5.5.1.2
Subtract from .
Step 5.5.5.2
Apply the distributive property.
Step 5.5.5.3
Simplify.
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Step 5.5.5.3.1
Multiply by by adding the exponents.
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Step 5.5.5.3.1.1
Move .
Step 5.5.5.3.1.2
Multiply by .
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Step 5.5.5.3.1.2.1
Raise to the power of .
Step 5.5.5.3.1.2.2
Use the power rule to combine exponents.
Step 5.5.5.3.1.3
Add and .
Step 5.5.5.3.2
Rewrite using the commutative property of multiplication.
Step 5.5.5.3.3
Multiply by .
Step 5.5.5.4
Simplify each term.
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Step 5.5.5.4.1
Multiply by by adding the exponents.
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Step 5.5.5.4.1.1
Move .
Step 5.5.5.4.1.2
Multiply by .
Step 5.5.5.4.2
Multiply by .
Step 5.6
Simplify the determinant.
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Step 5.6.1
Combine the opposite terms in .
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Step 5.6.1.1
Add and .
Step 5.6.1.2
Subtract from .
Step 5.6.1.3
Add and .
Step 5.6.2
Apply the distributive property.
Step 5.6.3
Simplify.
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Step 5.6.3.1
Rewrite using the commutative property of multiplication.
Step 5.6.3.2
Rewrite using the commutative property of multiplication.
Step 5.6.3.3
Rewrite using the commutative property of multiplication.
Step 5.6.4
Simplify each term.
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Step 5.6.4.1
Multiply by by adding the exponents.
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Step 5.6.4.1.1
Move .
Step 5.6.4.1.2
Multiply by .
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Step 5.6.4.1.2.1
Raise to the power of .
Step 5.6.4.1.2.2
Use the power rule to combine exponents.
Step 5.6.4.1.3
Add and .
Step 5.6.4.2
Multiply by .
Step 5.6.4.3
Multiply by .
Step 5.6.4.4
Multiply by by adding the exponents.
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Step 5.6.4.4.1
Move .
Step 5.6.4.4.2
Multiply by .
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Step 5.6.4.4.2.1
Raise to the power of .
Step 5.6.4.4.2.2
Use the power rule to combine exponents.
Step 5.6.4.4.3
Add and .
Step 5.6.4.5
Multiply by .
Step 5.6.4.6
Multiply by by adding the exponents.
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Step 5.6.4.6.1
Move .
Step 5.6.4.6.2
Multiply by .
Step 5.6.4.7
Multiply by .
Step 6
Set the characteristic polynomial equal to to find the eigenvalues .
Step 7
Solve for .
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Step 7.1
Factor the left side of the equation.
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Step 7.1.1
Factor out of .
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Step 7.1.1.1
Factor out of .
Step 7.1.1.2
Factor out of .
Step 7.1.1.3
Factor out of .
Step 7.1.1.4
Factor out of .
Step 7.1.1.5
Factor out of .
Step 7.1.2
Factor.
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Step 7.1.2.1
Factor using the AC method.
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Step 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 7.1.2.1.2
Write the factored form using these integers.
Step 7.1.2.2
Remove unnecessary parentheses.
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 .
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Step 7.3.1
Set equal to .
Step 7.3.2
Solve for .
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Step 7.3.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 7.3.2.2
Simplify .
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Step 7.3.2.2.1
Rewrite as .
Step 7.3.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 7.3.2.2.3
Plus or minus is .
Step 7.4
Set equal to and solve for .
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Step 7.4.1
Set equal to .
Step 7.4.2
Add to both sides of the equation.
Step 7.5
Set equal to and solve for .
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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.