Enter a problem...
Linear Algebra Examples
Step 1
The inverse of a matrix can be found using the formula where is the determinant.
Step 2
Step 2.1
The determinant of a matrix can be found using the formula .
Step 2.2
Simplify the determinant.
Step 2.2.1
Simplify each term.
Step 2.2.1.1
Expand using the FOIL Method.
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.
Step 2.2.1.2.1
Simplify each term.
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 .
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.
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.
Step 2.2.1.6.1
Simplify each term.
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 .
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
Step 6.1
Combine.
Step 6.2
Multiply by .
Step 6.3
Simplify the denominator.
Step 6.3.1
Expand using the FOIL Method.
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.
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.
Step 6.3.3.1
Rewrite as .
Step 6.3.3.2
Multiply by .
Step 6.3.4
Add and .
Step 7
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.
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
Step 9.1
Factor out of .
Step 9.2
Cancel the common factors.
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
Step 16.1
Expand using the FOIL Method.
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.
Step 16.2.1
Simplify each term.
Step 16.2.1.1
Cancel the common factor of .
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 .
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 .
Step 16.2.1.6.1
Combine and .
Step 16.2.1.6.2
Multiply by .
Step 16.2.1.7
Multiply .
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.
Step 16.3.1
Cancel the common factor of and .
Step 16.3.1.1
Factor out of .
Step 16.3.1.2
Cancel the common factors.
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 .
Step 16.3.3.1
Factor out of .
Step 16.3.3.2
Cancel the common factors.
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.
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.
Step 16.5.1
Simplify each term.
Step 16.5.1.1
Multiply .
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 .
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 .
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 .
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 .
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.
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 .
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.
Step 16.6.1
Cancel the common factor of and .
Step 16.6.1.1
Factor out of .
Step 16.6.1.2
Cancel the common factors.
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 .
Step 16.6.3.1
Factor out of .
Step 16.6.3.2
Cancel the common factors.
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.
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.
Step 16.8.1
Simplify each term.
Step 16.8.1.1
Multiply .
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 .
Step 16.8.1.3.1
Combine and .
Step 16.8.1.3.2
Multiply by .
Step 16.8.1.4
Multiply .
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 .
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 .
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.
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 .
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.
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.
Step 16.10.1
Simplify each term.
Step 16.10.1.1
Multiply .
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 .
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 .
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 .
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 .
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.
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 .
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.