Linear Algebra Examples

Find the Adjoint [[1,0,0],[0,2,6],[0,-4,-12]]
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
The determinant of a matrix can be found using the formula .
Step 2.1.2.2
Simplify the determinant.
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Step 2.1.2.2.1
Simplify each term.
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Step 2.1.2.2.1.1
Multiply by .
Step 2.1.2.2.1.2
Multiply .
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Step 2.1.2.2.1.2.1
Multiply by .
Step 2.1.2.2.1.2.2
Multiply by .
Step 2.1.2.2.2
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
The determinant of a matrix can be found using the formula .
Step 2.2.2.2
Simplify the determinant.
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Step 2.2.2.2.1
Simplify each term.
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Step 2.2.2.2.1.1
Multiply by .
Step 2.2.2.2.1.2
Multiply by .
Step 2.2.2.2.2
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
The determinant of a matrix can be found using the formula .
Step 2.3.2.2
Simplify the determinant.
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Step 2.3.2.2.1
Simplify each term.
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Step 2.3.2.2.1.1
Multiply by .
Step 2.3.2.2.1.2
Multiply by .
Step 2.3.2.2.2
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
The determinant of a matrix can be found using the formula .
Step 2.4.2.2
Simplify the determinant.
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Step 2.4.2.2.1
Simplify each term.
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Step 2.4.2.2.1.1
Multiply by .
Step 2.4.2.2.1.2
Multiply .
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Step 2.4.2.2.1.2.1
Multiply by .
Step 2.4.2.2.1.2.2
Multiply by .
Step 2.4.2.2.2
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
The determinant of a matrix can be found using the formula .
Step 2.5.2.2
Simplify the determinant.
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Step 2.5.2.2.1
Simplify each term.
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Step 2.5.2.2.1.1
Multiply by .
Step 2.5.2.2.1.2
Multiply by .
Step 2.5.2.2.2
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
The determinant of a matrix can be found using the formula .
Step 2.6.2.2
Simplify the determinant.
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Step 2.6.2.2.1
Simplify each term.
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Step 2.6.2.2.1.1
Multiply by .
Step 2.6.2.2.1.2
Multiply by .
Step 2.6.2.2.2
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
The determinant of a matrix can be found using the formula .
Step 2.7.2.2
Simplify the determinant.
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Step 2.7.2.2.1
Simplify each term.
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Step 2.7.2.2.1.1
Multiply by .
Step 2.7.2.2.1.2
Multiply by .
Step 2.7.2.2.2
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
The determinant of a matrix can be found using the formula .
Step 2.8.2.2
Simplify the determinant.
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Step 2.8.2.2.1
Simplify each term.
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Step 2.8.2.2.1.1
Multiply by .
Step 2.8.2.2.1.2
Multiply by .
Step 2.8.2.2.2
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
The determinant of a matrix can be found using the formula .
Step 2.9.2.2
Simplify the determinant.
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Step 2.9.2.2.1
Simplify each term.
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Step 2.9.2.2.1.1
Multiply by .
Step 2.9.2.2.1.2
Multiply by .
Step 2.9.2.2.2
Add and .
Step 2.10
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.