Linear Algebra Examples

Find the Adjoint [[2,0,1,4],[-2,1,0,0],[1,-1,3,2],[-1,0,0,2]]
Step 1
Consider the corresponding sign chart.
Step 2
Use the sign chart and the given matrix to find the cofactor of each element.
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Step 2.1
Calculate the minor for element .
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Step 2.1.1
The minor for is the determinant with row and column deleted.
Step 2.1.2
Evaluate the determinant.
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Step 2.1.2.1
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 2.1.2.1.1
Consider the corresponding sign chart.
Step 2.1.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.1.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.1.2.1.4
Multiply element by its cofactor.
Step 2.1.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.1.2.1.6
Multiply element by its cofactor.
Step 2.1.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.1.2.1.8
Multiply element by its cofactor.
Step 2.1.2.1.9
Add the terms together.
Step 2.1.2.2
Multiply by .
Step 2.1.2.3
Multiply by .
Step 2.1.2.4
Evaluate .
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Step 2.1.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.1.2.4.2
Simplify the determinant.
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Step 2.1.2.4.2.1
Simplify each term.
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Step 2.1.2.4.2.1.1
Multiply by .
Step 2.1.2.4.2.1.2
Multiply by .
Step 2.1.2.4.2.2
Add and .
Step 2.1.2.5
Simplify the determinant.
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Step 2.1.2.5.1
Multiply by .
Step 2.1.2.5.2
Add and .
Step 2.1.2.5.3
Add and .
Step 2.2
Calculate the minor for element .
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Step 2.2.1
The minor for is the determinant with row and column deleted.
Step 2.2.2
Evaluate the determinant.
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Step 2.2.2.1
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 2.2.2.1.1
Consider the corresponding sign chart.
Step 2.2.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.2.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.2.2.1.4
Multiply element by its cofactor.
Step 2.2.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.2.2.1.6
Multiply element by its cofactor.
Step 2.2.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.2.2.1.8
Multiply element by its cofactor.
Step 2.2.2.1.9
Add the terms together.
Step 2.2.2.2
Multiply by .
Step 2.2.2.3
Multiply by .
Step 2.2.2.4
Evaluate .
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Step 2.2.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.2.2.4.2
Simplify the determinant.
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Step 2.2.2.4.2.1
Simplify each term.
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Step 2.2.2.4.2.1.1
Multiply by .
Step 2.2.2.4.2.1.2
Multiply by .
Step 2.2.2.4.2.2
Add and .
Step 2.2.2.5
Simplify the determinant.
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Step 2.2.2.5.1
Multiply by .
Step 2.2.2.5.2
Add and .
Step 2.2.2.5.3
Add and .
Step 2.3
Calculate the minor for element .
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Step 2.3.1
The minor for is the determinant with row and column deleted.
Step 2.3.2
Evaluate the determinant.
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Step 2.3.2.1
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 2.3.2.1.1
Consider the corresponding sign chart.
Step 2.3.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.3.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.3.2.1.4
Multiply element by its cofactor.
Step 2.3.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.3.2.1.6
Multiply element by its cofactor.
Step 2.3.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.3.2.1.8
Multiply element by its cofactor.
Step 2.3.2.1.9
Add the terms together.
Step 2.3.2.2
Multiply by .
Step 2.3.2.3
Evaluate .
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Step 2.3.2.3.1
The determinant of a matrix can be found using the formula .
Step 2.3.2.3.2
Simplify the determinant.
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Step 2.3.2.3.2.1
Simplify each term.
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Step 2.3.2.3.2.1.1
Multiply by .
Step 2.3.2.3.2.1.2
Multiply by .
Step 2.3.2.3.2.2
Add and .
Step 2.3.2.4
Evaluate .
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Step 2.3.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.3.2.4.2
Simplify the determinant.
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Step 2.3.2.4.2.1
Simplify each term.
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Step 2.3.2.4.2.1.1
Multiply by .
Step 2.3.2.4.2.1.2
Multiply .
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Step 2.3.2.4.2.1.2.1
Multiply by .
Step 2.3.2.4.2.1.2.2
Multiply by .
Step 2.3.2.4.2.2
Add and .
Step 2.3.2.5
Simplify the determinant.
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Step 2.3.2.5.1
Simplify each term.
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Step 2.3.2.5.1.1
Multiply by .
Step 2.3.2.5.1.2
Multiply by .
Step 2.3.2.5.2
Subtract from .
Step 2.3.2.5.3
Add and .
Step 2.4
Calculate the minor for element .
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Step 2.4.1
The minor for is the determinant with row and column deleted.
Step 2.4.2
Evaluate the determinant.
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Step 2.4.2.1
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 2.4.2.1.1
Consider the corresponding sign chart.
Step 2.4.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.4.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.4.2.1.4
Multiply element by its cofactor.
Step 2.4.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.4.2.1.6
Multiply element by its cofactor.
Step 2.4.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.4.2.1.8
Multiply element by its cofactor.
Step 2.4.2.1.9
Add the terms together.
Step 2.4.2.2
Multiply by .
Step 2.4.2.3
Multiply by .
Step 2.4.2.4
Evaluate .
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Step 2.4.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.4.2.4.2
Simplify the determinant.
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Step 2.4.2.4.2.1
Simplify each term.
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Step 2.4.2.4.2.1.1
Multiply by .
Step 2.4.2.4.2.1.2
Multiply .
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Step 2.4.2.4.2.1.2.1
Multiply by .
Step 2.4.2.4.2.1.2.2
Multiply by .
Step 2.4.2.4.2.2
Add and .
Step 2.4.2.5
Simplify the determinant.
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Step 2.4.2.5.1
Multiply by .
Step 2.4.2.5.2
Add and .
Step 2.4.2.5.3
Add and .
Step 2.5
Calculate the minor for element .
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Step 2.5.1
The minor for is the determinant with row and column deleted.
Step 2.5.2
Evaluate the determinant.
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Step 2.5.2.1
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 2.5.2.1.1
Consider the corresponding sign chart.
Step 2.5.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.5.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.5.2.1.4
Multiply element by its cofactor.
Step 2.5.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.5.2.1.6
Multiply element by its cofactor.
Step 2.5.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.5.2.1.8
Multiply element by its cofactor.
Step 2.5.2.1.9
Add the terms together.
Step 2.5.2.2
Multiply by .
Step 2.5.2.3
Multiply by .
Step 2.5.2.4
Evaluate .
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Step 2.5.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.5.2.4.2
Simplify the determinant.
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Step 2.5.2.4.2.1
Simplify each term.
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Step 2.5.2.4.2.1.1
Multiply by .
Step 2.5.2.4.2.1.2
Multiply by .
Step 2.5.2.4.2.2
Add and .
Step 2.5.2.5
Simplify the determinant.
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Step 2.5.2.5.1
Multiply by .
Step 2.5.2.5.2
Add and .
Step 2.5.2.5.3
Add and .
Step 2.6
Calculate the minor for element .
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Step 2.6.1
The minor for is the determinant with row and column deleted.
Step 2.6.2
Evaluate the determinant.
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Step 2.6.2.1
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 2.6.2.1.1
Consider the corresponding sign chart.
Step 2.6.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.6.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.6.2.1.4
Multiply element by its cofactor.
Step 2.6.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.6.2.1.6
Multiply element by its cofactor.
Step 2.6.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.6.2.1.8
Multiply element by its cofactor.
Step 2.6.2.1.9
Add the terms together.
Step 2.6.2.2
Multiply by .
Step 2.6.2.3
Evaluate .
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Step 2.6.2.3.1
The determinant of a matrix can be found using the formula .
Step 2.6.2.3.2
Simplify the determinant.
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Step 2.6.2.3.2.1
Simplify each term.
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Step 2.6.2.3.2.1.1
Multiply by .
Step 2.6.2.3.2.1.2
Multiply .
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Step 2.6.2.3.2.1.2.1
Multiply by .
Step 2.6.2.3.2.1.2.2
Multiply by .
Step 2.6.2.3.2.2
Add and .
Step 2.6.2.4
Evaluate .
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Step 2.6.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.6.2.4.2
Simplify the determinant.
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Step 2.6.2.4.2.1
Simplify each term.
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Step 2.6.2.4.2.1.1
Multiply by .
Step 2.6.2.4.2.1.2
Multiply .
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Step 2.6.2.4.2.1.2.1
Multiply by .
Step 2.6.2.4.2.1.2.2
Multiply by .
Step 2.6.2.4.2.2
Add and .
Step 2.6.2.5
Simplify the determinant.
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Step 2.6.2.5.1
Simplify each term.
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Step 2.6.2.5.1.1
Multiply by .
Step 2.6.2.5.1.2
Multiply by .
Step 2.6.2.5.2
Add and .
Step 2.6.2.5.3
Add and .
Step 2.7
Calculate the minor for element .
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Step 2.7.1
The minor for is the determinant with row and column deleted.
Step 2.7.2
Evaluate the determinant.
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Step 2.7.2.1
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 2.7.2.1.1
Consider the corresponding sign chart.
Step 2.7.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.7.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.7.2.1.4
Multiply element by its cofactor.
Step 2.7.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.7.2.1.6
Multiply element by its cofactor.
Step 2.7.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.7.2.1.8
Multiply element by its cofactor.
Step 2.7.2.1.9
Add the terms together.
Step 2.7.2.2
Multiply by .
Step 2.7.2.3
Multiply by .
Step 2.7.2.4
Evaluate .
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Step 2.7.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.7.2.4.2
Simplify the determinant.
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Step 2.7.2.4.2.1
Simplify each term.
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Step 2.7.2.4.2.1.1
Multiply by .
Step 2.7.2.4.2.1.2
Multiply .
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Step 2.7.2.4.2.1.2.1
Multiply by .
Step 2.7.2.4.2.1.2.2
Multiply by .
Step 2.7.2.4.2.2
Add and .
Step 2.7.2.5
Simplify the determinant.
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Step 2.7.2.5.1
Multiply by .
Step 2.7.2.5.2
Subtract from .
Step 2.7.2.5.3
Add and .
Step 2.8
Calculate the minor for element .
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Step 2.8.1
The minor for is the determinant with row and column deleted.
Step 2.8.2
Evaluate the determinant.
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Step 2.8.2.1
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 2.8.2.1.1
Consider the corresponding sign chart.
Step 2.8.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.8.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.8.2.1.4
Multiply element by its cofactor.
Step 2.8.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.8.2.1.6
Multiply element by its cofactor.
Step 2.8.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.8.2.1.8
Multiply element by its cofactor.
Step 2.8.2.1.9
Add the terms together.
Step 2.8.2.2
Multiply by .
Step 2.8.2.3
Multiply by .
Step 2.8.2.4
Evaluate .
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Step 2.8.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.8.2.4.2
Simplify the determinant.
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Step 2.8.2.4.2.1
Simplify each term.
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Step 2.8.2.4.2.1.1
Multiply by .
Step 2.8.2.4.2.1.2
Multiply .
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Step 2.8.2.4.2.1.2.1
Multiply by .
Step 2.8.2.4.2.1.2.2
Multiply by .
Step 2.8.2.4.2.2
Add and .
Step 2.8.2.5
Simplify the determinant.
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Step 2.8.2.5.1
Multiply by .
Step 2.8.2.5.2
Subtract from .
Step 2.8.2.5.3
Add and .
Step 2.9
Calculate the minor for element .
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Step 2.9.1
The minor for is the determinant with row and column deleted.
Step 2.9.2
Evaluate the determinant.
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Step 2.9.2.1
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 2.9.2.1.1
Consider the corresponding sign chart.
Step 2.9.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.9.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.9.2.1.4
Multiply element by its cofactor.
Step 2.9.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.9.2.1.6
Multiply element by its cofactor.
Step 2.9.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.9.2.1.8
Multiply element by its cofactor.
Step 2.9.2.1.9
Add the terms together.
Step 2.9.2.2
Multiply by .
Step 2.9.2.3
Multiply by .
Step 2.9.2.4
Evaluate .
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Step 2.9.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.9.2.4.2
Simplify the determinant.
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Step 2.9.2.4.2.1
Simplify each term.
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Step 2.9.2.4.2.1.1
Multiply by .
Step 2.9.2.4.2.1.2
Multiply by .
Step 2.9.2.4.2.2
Add and .
Step 2.9.2.5
Simplify the determinant.
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Step 2.9.2.5.1
Multiply by .
Step 2.9.2.5.2
Subtract from .
Step 2.9.2.5.3
Add and .
Step 2.10
Calculate the minor for element .
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Step 2.10.1
The minor for is the determinant with row and column deleted.
Step 2.10.2
Evaluate the determinant.
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Step 2.10.2.1
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 2.10.2.1.1
Consider the corresponding sign chart.
Step 2.10.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.10.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.10.2.1.4
Multiply element by its cofactor.
Step 2.10.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.10.2.1.6
Multiply element by its cofactor.
Step 2.10.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.10.2.1.8
Multiply element by its cofactor.
Step 2.10.2.1.9
Add the terms together.
Step 2.10.2.2
Multiply by .
Step 2.10.2.3
Multiply by .
Step 2.10.2.4
Evaluate .
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Step 2.10.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.10.2.4.2
Simplify the determinant.
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Step 2.10.2.4.2.1
Simplify each term.
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Step 2.10.2.4.2.1.1
Multiply by .
Step 2.10.2.4.2.1.2
Multiply by .
Step 2.10.2.4.2.2
Add and .
Step 2.10.2.5
Simplify the determinant.
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Step 2.10.2.5.1
Multiply by .
Step 2.10.2.5.2
Add and .
Step 2.10.2.5.3
Add and .
Step 2.11
Calculate the minor for element .
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Step 2.11.1
The minor for is the determinant with row and column deleted.
Step 2.11.2
Evaluate the determinant.
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Step 2.11.2.1
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 2.11.2.1.1
Consider the corresponding sign chart.
Step 2.11.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.11.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.11.2.1.4
Multiply element by its cofactor.
Step 2.11.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.11.2.1.6
Multiply element by its cofactor.
Step 2.11.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.11.2.1.8
Multiply element by its cofactor.
Step 2.11.2.1.9
Add the terms together.
Step 2.11.2.2
Multiply by .
Step 2.11.2.3
Multiply by .
Step 2.11.2.4
Evaluate .
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Step 2.11.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.11.2.4.2
Simplify the determinant.
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Step 2.11.2.4.2.1
Simplify each term.
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Step 2.11.2.4.2.1.1
Multiply by .
Step 2.11.2.4.2.1.2
Multiply .
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Step 2.11.2.4.2.1.2.1
Multiply by .
Step 2.11.2.4.2.1.2.2
Multiply by .
Step 2.11.2.4.2.2
Add and .
Step 2.11.2.5
Simplify the determinant.
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Step 2.11.2.5.1
Multiply by .
Step 2.11.2.5.2
Add and .
Step 2.11.2.5.3
Add and .
Step 2.12
Calculate the minor for element .
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Step 2.12.1
The minor for is the determinant with row and column deleted.
Step 2.12.2
Evaluate the determinant.
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Step 2.12.2.1
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 2.12.2.1.1
Consider the corresponding sign chart.
Step 2.12.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.12.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.12.2.1.4
Multiply element by its cofactor.
Step 2.12.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.12.2.1.6
Multiply element by its cofactor.
Step 2.12.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.12.2.1.8
Multiply element by its cofactor.
Step 2.12.2.1.9
Add the terms together.
Step 2.12.2.2
Multiply by .
Step 2.12.2.3
Multiply by .
Step 2.12.2.4
Evaluate .
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Step 2.12.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.12.2.4.2
Simplify the determinant.
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Step 2.12.2.4.2.1
Simplify each term.
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Step 2.12.2.4.2.1.1
Multiply by .
Step 2.12.2.4.2.1.2
Multiply .
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Step 2.12.2.4.2.1.2.1
Multiply by .
Step 2.12.2.4.2.1.2.2
Multiply by .
Step 2.12.2.4.2.2
Add and .
Step 2.12.2.5
Simplify the determinant.
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Step 2.12.2.5.1
Multiply by .
Step 2.12.2.5.2
Add and .
Step 2.12.2.5.3
Add and .
Step 2.13
Calculate the minor for element .
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Step 2.13.1
The minor for is the determinant with row and column deleted.
Step 2.13.2
Evaluate the determinant.
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Step 2.13.2.1
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 2.13.2.1.1
Consider the corresponding sign chart.
Step 2.13.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.13.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.13.2.1.4
Multiply element by its cofactor.
Step 2.13.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.13.2.1.6
Multiply element by its cofactor.
Step 2.13.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.13.2.1.8
Multiply element by its cofactor.
Step 2.13.2.1.9
Add the terms together.
Step 2.13.2.2
Multiply by .
Step 2.13.2.3
Multiply by .
Step 2.13.2.4
Evaluate .
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Step 2.13.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.13.2.4.2
Simplify the determinant.
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Step 2.13.2.4.2.1
Simplify each term.
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Step 2.13.2.4.2.1.1
Multiply by .
Step 2.13.2.4.2.1.2
Multiply by .
Step 2.13.2.4.2.2
Subtract from .
Step 2.13.2.5
Simplify the determinant.
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Step 2.13.2.5.1
Multiply by .
Step 2.13.2.5.2
Add and .
Step 2.13.2.5.3
Add and .
Step 2.14
Calculate the minor for element .
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Step 2.14.1
The minor for is the determinant with row and column deleted.
Step 2.14.2
Evaluate the determinant.
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Step 2.14.2.1
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 2.14.2.1.1
Consider the corresponding sign chart.
Step 2.14.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.14.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.14.2.1.4
Multiply element by its cofactor.
Step 2.14.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.14.2.1.6
Multiply element by its cofactor.
Step 2.14.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.14.2.1.8
Multiply element by its cofactor.
Step 2.14.2.1.9
Add the terms together.
Step 2.14.2.2
Multiply by .
Step 2.14.2.3
Multiply by .
Step 2.14.2.4
Evaluate .
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Step 2.14.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.14.2.4.2
Simplify the determinant.
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Step 2.14.2.4.2.1
Simplify each term.
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Step 2.14.2.4.2.1.1
Multiply by .
Step 2.14.2.4.2.1.2
Multiply by .
Step 2.14.2.4.2.2
Subtract from .
Step 2.14.2.5
Simplify the determinant.
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Step 2.14.2.5.1
Multiply by .
Step 2.14.2.5.2
Add and .
Step 2.14.2.5.3
Add and .
Step 2.15
Calculate the minor for element .
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Step 2.15.1
The minor for is the determinant with row and column deleted.
Step 2.15.2
Evaluate the determinant.
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Step 2.15.2.1
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 2.15.2.1.1
Consider the corresponding sign chart.
Step 2.15.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.15.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.15.2.1.4
Multiply element by its cofactor.
Step 2.15.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.15.2.1.6
Multiply element by its cofactor.
Step 2.15.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.15.2.1.8
Multiply element by its cofactor.
Step 2.15.2.1.9
Add the terms together.
Step 2.15.2.2
Multiply by .
Step 2.15.2.3
Evaluate .
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Step 2.15.2.3.1
The determinant of a matrix can be found using the formula .
Step 2.15.2.3.2
Simplify the determinant.
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Step 2.15.2.3.2.1
Simplify each term.
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Step 2.15.2.3.2.1.1
Multiply by .
Step 2.15.2.3.2.1.2
Multiply .
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Step 2.15.2.3.2.1.2.1
Multiply by .
Step 2.15.2.3.2.1.2.2
Multiply by .
Step 2.15.2.3.2.2
Add and .
Step 2.15.2.4
Evaluate .
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Step 2.15.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.15.2.4.2
Simplify the determinant.
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Step 2.15.2.4.2.1
Simplify each term.
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Step 2.15.2.4.2.1.1
Multiply by .
Step 2.15.2.4.2.1.2
Multiply by .
Step 2.15.2.4.2.2
Subtract from .
Step 2.15.2.5
Simplify the determinant.
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Step 2.15.2.5.1
Simplify each term.
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Step 2.15.2.5.1.1
Multiply by .
Step 2.15.2.5.1.2
Multiply by .
Step 2.15.2.5.2
Add and .
Step 2.15.2.5.3
Add and .
Step 2.16
Calculate the minor for element .
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Step 2.16.1
The minor for is the determinant with row and column deleted.
Step 2.16.2
Evaluate the determinant.
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Step 2.16.2.1
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 2.16.2.1.1
Consider the corresponding sign chart.
Step 2.16.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.16.2.1.3
The minor for is the determinant with row and column deleted.
Step 2.16.2.1.4
Multiply element by its cofactor.
Step 2.16.2.1.5
The minor for is the determinant with row and column deleted.
Step 2.16.2.1.6
Multiply element by its cofactor.
Step 2.16.2.1.7
The minor for is the determinant with row and column deleted.
Step 2.16.2.1.8
Multiply element by its cofactor.
Step 2.16.2.1.9
Add the terms together.
Step 2.16.2.2
Multiply by .
Step 2.16.2.3
Evaluate .
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Step 2.16.2.3.1
The determinant of a matrix can be found using the formula .
Step 2.16.2.3.2
Simplify the determinant.
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Step 2.16.2.3.2.1
Simplify each term.
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Step 2.16.2.3.2.1.1
Multiply by .
Step 2.16.2.3.2.1.2
Multiply .
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Step 2.16.2.3.2.1.2.1
Multiply by .
Step 2.16.2.3.2.1.2.2
Multiply by .
Step 2.16.2.3.2.2
Add and .
Step 2.16.2.4
Evaluate .
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Step 2.16.2.4.1
The determinant of a matrix can be found using the formula .
Step 2.16.2.4.2
Simplify the determinant.
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Step 2.16.2.4.2.1
Simplify each term.
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Step 2.16.2.4.2.1.1
Multiply by .
Step 2.16.2.4.2.1.2
Multiply by .
Step 2.16.2.4.2.2
Subtract from .
Step 2.16.2.5
Simplify the determinant.
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Step 2.16.2.5.1
Simplify each term.
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Step 2.16.2.5.1.1
Multiply by .
Step 2.16.2.5.1.2
Multiply by .
Step 2.16.2.5.2
Add and .
Step 2.16.2.5.3
Add and .
Step 2.17
The cofactor matrix is a matrix of the minors with the sign changed for the elements in the positions on the sign chart.
Step 3
Transpose the matrix by switching its rows to columns.