Linear Algebra Examples

Find the Inverse A=[[-3-3i,1-3i],[-3+i,2+4i]]
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
Expand using the FOIL Method.
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Step 2.2.1.1.1
Apply the distributive property.
Step 2.2.1.1.2
Apply the distributive property.
Step 2.2.1.1.3
Apply the distributive property.
Step 2.2.1.2
Simplify and combine like terms.
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Step 2.2.1.2.1
Simplify each term.
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Step 2.2.1.2.1.1
Multiply by .
Step 2.2.1.2.1.2
Multiply by .
Step 2.2.1.2.1.3
Multiply by .
Step 2.2.1.2.1.4
Multiply .
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Step 2.2.1.2.1.4.1
Multiply by .
Step 2.2.1.2.1.4.2
Raise to the power of .
Step 2.2.1.2.1.4.3
Raise to the power of .
Step 2.2.1.2.1.4.4
Use the power rule to combine exponents.
Step 2.2.1.2.1.4.5
Add and .
Step 2.2.1.2.1.5
Rewrite as .
Step 2.2.1.2.1.6
Multiply by .
Step 2.2.1.2.2
Add and .
Step 2.2.1.2.3
Subtract from .
Step 2.2.1.3
Apply the distributive property.
Step 2.2.1.4
Multiply by .
Step 2.2.1.5
Expand using the FOIL Method.
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Step 2.2.1.5.1
Apply the distributive property.
Step 2.2.1.5.2
Apply the distributive property.
Step 2.2.1.5.3
Apply the distributive property.
Step 2.2.1.6
Simplify and combine like terms.
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Step 2.2.1.6.1
Simplify each term.
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Step 2.2.1.6.1.1
Multiply by .
Step 2.2.1.6.1.2
Multiply by .
Step 2.2.1.6.1.3
Multiply by .
Step 2.2.1.6.1.4
Multiply .
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Step 2.2.1.6.1.4.1
Multiply by .
Step 2.2.1.6.1.4.2
Raise to the power of .
Step 2.2.1.6.1.4.3
Raise to the power of .
Step 2.2.1.6.1.4.4
Use the power rule to combine exponents.
Step 2.2.1.6.1.4.5
Add and .
Step 2.2.1.6.1.5
Rewrite as .
Step 2.2.1.6.1.6
Multiply by .
Step 2.2.1.6.2
Subtract from .
Step 2.2.1.6.3
Subtract from .
Step 2.2.1.6.4
Subtract from .
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
Multiply the numerator and denominator of by the conjugate of to make the denominator real.
Step 6
Multiply.
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Step 6.1
Combine.
Step 6.2
Multiply by .
Step 6.3
Simplify the denominator.
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Step 6.3.1
Expand using the FOIL Method.
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Step 6.3.1.1
Apply the distributive property.
Step 6.3.1.2
Apply the distributive property.
Step 6.3.1.3
Apply the distributive property.
Step 6.3.2
Simplify.
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Step 6.3.2.1
Multiply by .
Step 6.3.2.2
Multiply by .
Step 6.3.2.3
Multiply by .
Step 6.3.2.4
Multiply by .
Step 6.3.2.5
Raise to the power of .
Step 6.3.2.6
Raise to the power of .
Step 6.3.2.7
Use the power rule to combine exponents.
Step 6.3.2.8
Add and .
Step 6.3.2.9
Subtract from .
Step 6.3.2.10
Add and .
Step 6.3.3
Simplify each term.
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Step 6.3.3.1
Rewrite as .
Step 6.3.3.2
Multiply by .
Step 6.3.4
Add and .
Step 7
Cancel the common factor of and .
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Step 7.1
Factor out of .
Step 7.2
Factor out of .
Step 7.3
Factor out of .
Step 7.4
Cancel the common factors.
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Step 7.4.1
Factor out of .
Step 7.4.2
Cancel the common factor.
Step 7.4.3
Rewrite the expression.
Step 8
Split the fraction into two fractions.
Step 9
Cancel the common factor of and .
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Step 9.1
Factor out of .
Step 9.2
Cancel the common factors.
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Step 9.2.1
Factor out of .
Step 9.2.2
Cancel the common factor.
Step 9.2.3
Rewrite the expression.
Step 10
Apply the distributive property.
Step 11
Multiply by .
Step 12
Multiply by .
Step 13
Apply the distributive property.
Step 14
Multiply by .
Step 15
Multiply by each element of the matrix.
Step 16
Simplify each element in the matrix.
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Step 16.1
Expand using the FOIL Method.
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Step 16.1.1
Apply the distributive property.
Step 16.1.2
Apply the distributive property.
Step 16.1.3
Apply the distributive property.
Step 16.2
Simplify and combine like terms.
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Step 16.2.1
Simplify each term.
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Step 16.2.1.1
Cancel the common factor of .
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Step 16.2.1.1.1
Factor out of .
Step 16.2.1.1.2
Cancel the common factor.
Step 16.2.1.1.3
Rewrite the expression.
Step 16.2.1.2
Cancel the common factor of .
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Step 16.2.1.2.1
Factor out of .
Step 16.2.1.2.2
Factor out of .
Step 16.2.1.2.3
Cancel the common factor.
Step 16.2.1.2.4
Rewrite the expression.
Step 16.2.1.3
Combine and .
Step 16.2.1.4
Multiply by .
Step 16.2.1.5
Combine and .
Step 16.2.1.6
Multiply .
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Step 16.2.1.6.1
Combine and .
Step 16.2.1.6.2
Multiply by .
Step 16.2.1.7
Multiply .
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Step 16.2.1.7.1
Combine and .
Step 16.2.1.7.2
Multiply by .
Step 16.2.1.7.3
Combine and .
Step 16.2.1.7.4
Raise to the power of .
Step 16.2.1.7.5
Raise to the power of .
Step 16.2.1.7.6
Use the power rule to combine exponents.
Step 16.2.1.7.7
Add and .
Step 16.2.1.8
Rewrite as .
Step 16.2.1.9
Multiply by .
Step 16.2.1.10
Move the negative in front of the fraction.
Step 16.2.2
Combine the numerators over the common denominator.
Step 16.2.3
Subtract from .
Step 16.2.4
Combine the numerators over the common denominator.
Step 16.3
Simplify each term.
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Step 16.3.1
Cancel the common factor of and .
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Step 16.3.1.1
Factor out of .
Step 16.3.1.2
Cancel the common factors.
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Step 16.3.1.2.1
Factor out of .
Step 16.3.1.2.2
Cancel the common factor.
Step 16.3.1.2.3
Rewrite the expression.
Step 16.3.2
Move the negative in front of the fraction.
Step 16.3.3
Cancel the common factor of and .
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Step 16.3.3.1
Factor out of .
Step 16.3.3.2
Cancel the common factors.
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Step 16.3.3.2.1
Factor out of .
Step 16.3.3.2.2
Cancel the common factor.
Step 16.3.3.2.3
Rewrite the expression.
Step 16.4
Expand using the FOIL Method.
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Step 16.4.1
Apply the distributive property.
Step 16.4.2
Apply the distributive property.
Step 16.4.3
Apply the distributive property.
Step 16.5
Simplify and combine like terms.
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Step 16.5.1
Simplify each term.
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Step 16.5.1.1
Multiply .
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Step 16.5.1.1.1
Combine and .
Step 16.5.1.1.2
Multiply by .
Step 16.5.1.2
Move the negative in front of the fraction.
Step 16.5.1.3
Multiply .
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Step 16.5.1.3.1
Combine and .
Step 16.5.1.3.2
Multiply by .
Step 16.5.1.3.3
Combine and .
Step 16.5.1.4
Multiply .
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Step 16.5.1.4.1
Combine and .
Step 16.5.1.4.2
Multiply by .
Step 16.5.1.5
Move the negative in front of the fraction.
Step 16.5.1.6
Multiply .
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Step 16.5.1.6.1
Combine and .
Step 16.5.1.6.2
Multiply by .
Step 16.5.1.6.3
Combine and .
Step 16.5.1.6.4
Raise to the power of .
Step 16.5.1.6.5
Raise to the power of .
Step 16.5.1.6.6
Use the power rule to combine exponents.
Step 16.5.1.6.7
Add and .
Step 16.5.1.7
Rewrite as .
Step 16.5.1.8
Multiply by .
Step 16.5.1.9
Move the negative in front of the fraction.
Step 16.5.2
To write as a fraction with a common denominator, multiply by .
Step 16.5.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 16.5.3.1
Multiply by .
Step 16.5.3.2
Multiply by .
Step 16.5.4
Combine the numerators over the common denominator.
Step 16.5.5
Simplify the numerator.
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Step 16.5.5.1
Multiply by .
Step 16.5.5.2
Subtract from .
Step 16.5.6
To write as a fraction with a common denominator, multiply by .
Step 16.5.7
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 16.5.7.1
Multiply by .
Step 16.5.7.2
Multiply by .
Step 16.5.8
Combine the numerators over the common denominator.
Step 16.6
Simplify each term.
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Step 16.6.1
Cancel the common factor of and .
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Step 16.6.1.1
Factor out of .
Step 16.6.1.2
Cancel the common factors.
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Step 16.6.1.2.1
Factor out of .
Step 16.6.1.2.2
Cancel the common factor.
Step 16.6.1.2.3
Rewrite the expression.
Step 16.6.2
Move the negative in front of the fraction.
Step 16.6.3
Cancel the common factor of and .
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Step 16.6.3.1
Factor out of .
Step 16.6.3.2
Cancel the common factors.
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Step 16.6.3.2.1
Factor out of .
Step 16.6.3.2.2
Cancel the common factor.
Step 16.6.3.2.3
Rewrite the expression.
Step 16.6.4
Move the negative in front of the fraction.
Step 16.7
Expand using the FOIL Method.
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Step 16.7.1
Apply the distributive property.
Step 16.7.2
Apply the distributive property.
Step 16.7.3
Apply the distributive property.
Step 16.8
Simplify and combine like terms.
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Step 16.8.1
Simplify each term.
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Step 16.8.1.1
Multiply .
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Step 16.8.1.1.1
Combine and .
Step 16.8.1.1.2
Multiply by .
Step 16.8.1.2
Combine and .
Step 16.8.1.3
Multiply .
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Step 16.8.1.3.1
Combine and .
Step 16.8.1.3.2
Multiply by .
Step 16.8.1.4
Multiply .
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Step 16.8.1.4.1
Combine and .
Step 16.8.1.4.2
Raise to the power of .
Step 16.8.1.4.3
Raise to the power of .
Step 16.8.1.4.4
Use the power rule to combine exponents.
Step 16.8.1.4.5
Add and .
Step 16.8.1.5
Rewrite as .
Step 16.8.1.6
Multiply by .
Step 16.8.1.7
Move the negative in front of the fraction.
Step 16.8.1.8
Multiply .
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Step 16.8.1.8.1
Multiply by .
Step 16.8.1.8.2
Multiply by .
Step 16.8.2
To write as a fraction with a common denominator, multiply by .
Step 16.8.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 16.8.3.1
Multiply by .
Step 16.8.3.2
Multiply by .
Step 16.8.4
Combine the numerators over the common denominator.
Step 16.8.5
Simplify the numerator.
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Step 16.8.5.1
Multiply by .
Step 16.8.5.2
Add and .
Step 16.8.6
To write as a fraction with a common denominator, multiply by .
Step 16.8.7
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 16.8.7.1
Multiply by .
Step 16.8.7.2
Multiply by .
Step 16.8.8
Combine the numerators over the common denominator.
Step 16.9
Expand using the FOIL Method.
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Step 16.9.1
Apply the distributive property.
Step 16.9.2
Apply the distributive property.
Step 16.9.3
Apply the distributive property.
Step 16.10
Simplify and combine like terms.
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Step 16.10.1
Simplify each term.
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Step 16.10.1.1
Multiply .
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Step 16.10.1.1.1
Combine and .
Step 16.10.1.1.2
Multiply by .
Step 16.10.1.2
Move the negative in front of the fraction.
Step 16.10.1.3
Multiply .
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Step 16.10.1.3.1
Combine and .
Step 16.10.1.3.2
Multiply by .
Step 16.10.1.3.3
Combine and .
Step 16.10.1.4
Move the negative in front of the fraction.
Step 16.10.1.5
Multiply .
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Step 16.10.1.5.1
Combine and .
Step 16.10.1.5.2
Multiply by .
Step 16.10.1.6
Move the negative in front of the fraction.
Step 16.10.1.7
Multiply .
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Step 16.10.1.7.1
Combine and .
Step 16.10.1.7.2
Multiply by .
Step 16.10.1.7.3
Combine and .
Step 16.10.1.7.4
Raise to the power of .
Step 16.10.1.7.5
Raise to the power of .
Step 16.10.1.7.6
Use the power rule to combine exponents.
Step 16.10.1.7.7
Add and .
Step 16.10.1.8
Rewrite as .
Step 16.10.1.9
Multiply by .
Step 16.10.2
To write as a fraction with a common denominator, multiply by .
Step 16.10.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 16.10.3.1
Multiply by .
Step 16.10.3.2
Multiply by .
Step 16.10.4
Combine the numerators over the common denominator.
Step 16.10.5
Simplify the numerator.
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Step 16.10.5.1
Multiply by .
Step 16.10.5.2
Add and .
Step 16.10.6
To write as a fraction with a common denominator, multiply by .
Step 16.10.7
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 16.10.7.1
Multiply by .
Step 16.10.7.2
Multiply by .
Step 16.10.8
Combine the numerators over the common denominator.
Step 16.11
Move the negative in front of the fraction.