# Algebra Examples

Step 1
The inverse of a matrix can be found using the formula where is the determinant.
Step 2
Find the determinant.
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
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.
Step 7.1
Cancel the common factor of .
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
Multiply .
Step 7.3.1
Multiply by .
Step 7.3.2
Combine and .
Step 7.4
Cancel the common factor of .
Step 7.4.1
Move the leading negative in into the numerator.
Step 7.4.2
Factor out of .
Step 7.4.3
Factor out of .
Step 7.4.4
Cancel the common factor.
Step 7.4.5
Rewrite the expression.
Step 7.5
Combine and .
Step 7.6
Multiply by .
Step 7.7
Multiply by .