Linear Algebra Examples

Find Pivot Positions and Pivot Columns [[a,a,1-a,1],[a,a^2a,1-a^2,1],[2a,a+a^2,2-2a,a+2]]
Step 1
Find the reduced row echelon form.
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Step 1.1
Multiply by by adding the exponents.
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Step 1.1.1
Multiply by .
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Step 1.1.1.1
Raise to the power of .
Step 1.1.1.2
Use the power rule to combine exponents.
Step 1.1.2
Add and .
Step 1.2
Multiply each element of by to make the entry at a .
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Step 1.2.1
Multiply each element of by to make the entry at a .
Step 1.2.2
Simplify .
Step 1.3
Perform the row operation to make the entry at a .
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Step 1.3.1
Perform the row operation to make the entry at a .
Step 1.3.2
Simplify .
Step 1.4
Perform the row operation to make the entry at a .
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Step 1.4.1
Perform the row operation to make the entry at a .
Step 1.4.2
Simplify .
Step 1.5
Multiply each element of by to make the entry at a .
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Step 1.5.1
Multiply each element of by to make the entry at a .
Step 1.5.2
Simplify .
Step 1.6
Perform the row operation to make the entry at a .
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Step 1.6.1
Perform the row operation to make the entry at a .
Step 1.6.2
Simplify .
Step 1.7
Multiply each element of by to make the entry at a .
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Step 1.7.1
Multiply each element of by to make the entry at a .
Step 1.7.2
Simplify .
Step 1.8
Perform the row operation to make the entry at a .
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Step 1.8.1
Perform the row operation to make the entry at a .
Step 1.8.2
Simplify .
Step 1.9
Perform the row operation to make the entry at a .
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Step 1.9.1
Perform the row operation to make the entry at a .
Step 1.9.2
Simplify .
Step 1.10
Perform the row operation to make the entry at a .
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Step 1.10.1
Perform the row operation to make the entry at a .
Step 1.10.2
Simplify .
Step 2
The pivot positions are the locations with the leading in each row. The pivot columns are the columns that have a pivot position.
Pivot Positions: and
Pivot Columns: and