Linear Algebra Examples

Find the Inverse [[22000,45000],[18000,27000]]
Step 1
The inverse of a matrix can be found using the formula where is the determinant.
Step 2
Find the determinant.
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Step 2.1
The determinant of a matrix can be found using the formula .
Step 2.2
Simplify the determinant.
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Step 2.2.1
Simplify each term.
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Step 2.2.1.1
Multiply by .
Step 2.2.1.2
Multiply by .
Step 2.2.2
Subtract from .
Step 3
Since the determinant is non-zero, the inverse exists.
Step 4
Substitute the known values into the formula for the inverse.
Step 5
Move the negative in front of the fraction.
Step 6
Multiply by each element of the matrix.
Step 7
Simplify each element in the matrix.
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Step 7.1
Cancel the common factor of .
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Step 7.1.1
Move the leading negative in into the numerator.
Step 7.1.2
Factor out of .
Step 7.1.3
Cancel the common factor.
Step 7.1.4
Rewrite the expression.
Step 7.2
Move the negative in front of the fraction.
Step 7.3
Cancel the common factor of .
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Step 7.3.1
Move the leading negative in into the numerator.
Step 7.3.2
Factor out of .
Step 7.3.3
Factor out of .
Step 7.3.4
Cancel the common factor.
Step 7.3.5
Rewrite the expression.
Step 7.4
Combine and .
Step 7.5
Multiply by .
Step 7.6
Cancel the common factor of .
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Step 7.6.1
Move the leading negative in into the numerator.
Step 7.6.2
Factor out of .
Step 7.6.3
Factor out of .
Step 7.6.4
Cancel the common factor.
Step 7.6.5
Rewrite the expression.
Step 7.7
Combine and .
Step 7.8
Multiply by .
Step 7.9
Cancel the common factor of .
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Step 7.9.1
Move the leading negative in into the numerator.
Step 7.9.2
Factor out of .
Step 7.9.3
Factor out of .
Step 7.9.4
Cancel the common factor.
Step 7.9.5
Rewrite the expression.
Step 7.10
Combine and .
Step 7.11
Multiply by .
Step 7.12
Move the negative in front of the fraction.