Linear Algebra Examples

Find the Determinant [[1,x,x^2],[1,x^2,x^4],[1,x^4,x^6]]
Step 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.
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Step 1.1
Consider the corresponding sign chart.
Step 1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 1.3
The minor for is the determinant with row and column deleted.
Step 1.4
Multiply element by its cofactor.
Step 1.5
The minor for is the determinant with row and column deleted.
Step 1.6
Multiply element by its cofactor.
Step 1.7
The minor for is the determinant with row and column deleted.
Step 1.8
Multiply element by its cofactor.
Step 1.9
Add the terms together.
Step 2
Evaluate .
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Step 2.1
The determinant of a matrix can be found using the formula .
Step 2.2
Multiply by .
Step 3
Evaluate .
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Step 3.1
The determinant of a matrix can be found using the formula .
Step 3.2
Multiply by .
Step 4
Evaluate .
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Step 4.1
The determinant of a matrix can be found using the formula .
Step 4.2
Multiply by .
Step 5
Simplify the determinant.
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Step 5.1
Simplify each term.
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Step 5.1.1
Apply the distributive property.
Step 5.1.2
Multiply by by adding the exponents.
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Step 5.1.2.1
Move .
Step 5.1.2.2
Multiply by .
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Step 5.1.2.2.1
Raise to the power of .
Step 5.1.2.2.2
Use the power rule to combine exponents.
Step 5.1.2.3
Add and .
Step 5.1.3
Rewrite using the commutative property of multiplication.
Step 5.1.4
Simplify each term.
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Step 5.1.4.1
Multiply by by adding the exponents.
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Step 5.1.4.1.1
Move .
Step 5.1.4.1.2
Multiply by .
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Step 5.1.4.1.2.1
Raise to the power of .
Step 5.1.4.1.2.2
Use the power rule to combine exponents.
Step 5.1.4.1.3
Add and .
Step 5.1.4.2
Multiply by .
Step 5.1.4.3
Multiply by .
Step 5.1.5
Apply the distributive property.
Step 5.1.6
Multiply by by adding the exponents.
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Step 5.1.6.1
Use the power rule to combine exponents.
Step 5.1.6.2
Add and .
Step 5.1.7
Rewrite using the commutative property of multiplication.
Step 5.1.8
Multiply by by adding the exponents.
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Step 5.1.8.1
Move .
Step 5.1.8.2
Use the power rule to combine exponents.
Step 5.1.8.3
Add and .
Step 5.1.9
Apply the distributive property.
Step 5.1.10
Multiply by by adding the exponents.
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Step 5.1.10.1
Move .
Step 5.1.10.2
Use the power rule to combine exponents.
Step 5.1.10.3
Add and .
Step 5.1.11
Rewrite using the commutative property of multiplication.
Step 5.1.12
Simplify each term.
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Step 5.1.12.1
Multiply by by adding the exponents.
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Step 5.1.12.1.1
Move .
Step 5.1.12.1.2
Use the power rule to combine exponents.
Step 5.1.12.1.3
Add and .
Step 5.1.12.2
Multiply by .
Step 5.1.12.3
Multiply by .
Step 5.2
Combine the opposite terms in .
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Step 5.2.1
Subtract from .
Step 5.2.2
Add and .