Linear Algebra Examples

Find the Inverse [[10,9],[-6,-5]]
[109-6-5]
Step 1
The inverse of a 2×2 matrix can be found using the formula 1ad-bc[d-b-ca] where ad-bc is the determinant.
Step 2
Find the determinant.
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Step 2.1
The determinant of a 2×2 matrix can be found using the formula |abcd|=ad-cb.
10-5-(-69)
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 10 by -5.
-50-(-69)
Step 2.2.1.2
Multiply -(-69).
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Step 2.2.1.2.1
Multiply -6 by 9.
-50--54
Step 2.2.1.2.2
Multiply -1 by -54.
-50+54
-50+54
-50+54
Step 2.2.2
Add -50 and 54.
4
4
4
Step 3
Since the determinant is non-zero, the inverse exists.
Step 4
Substitute the known values into the formula for the inverse.
14[-5-9610]
Step 5
Multiply 14 by each element of the matrix.
[14-514-91461410]
Step 6
Simplify each element in the matrix.
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Step 6.1
Combine 14 and -5.
[-5414-91461410]
Step 6.2
Move the negative in front of the fraction.
[-5414-91461410]
Step 6.3
Combine 14 and -9.
[-54-941461410]
Step 6.4
Move the negative in front of the fraction.
[-54-941461410]
Step 6.5
Cancel the common factor of 2.
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Step 6.5.1
Factor 2 out of 4.
[-54-9412(2)61410]
Step 6.5.2
Factor 2 out of 6.
[-54-94122(23)1410]
Step 6.5.3
Cancel the common factor.
[-54-94122(23)1410]
Step 6.5.4
Rewrite the expression.
[-54-941231410]
[-54-941231410]
Step 6.6
Combine 12 and 3.
[-54-94321410]
Step 6.7
Cancel the common factor of 2.
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Step 6.7.1
Factor 2 out of 4.
[-54-943212(2)10]
Step 6.7.2
Factor 2 out of 10.
[-54-9432122(25)]
Step 6.7.3
Cancel the common factor.
[-54-9432122(25)]
Step 6.7.4
Rewrite the expression.
[-54-9432125]
[-54-9432125]
Step 6.8
Combine 12 and 5.
[-54-943252]
[-54-943252]
 [x2  12  π  xdx ]