# Linear 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
Multiply by each element of the matrix.
Step 6
Simplify each element in the matrix.
Step 6.1
Combine and .
Step 6.2
Cancel the common factor of .
Step 6.2.1
Factor out of .
Step 6.2.2
Factor out of .
Step 6.2.3
Cancel the common factor.
Step 6.2.4
Rewrite the expression.
Step 6.3
Combine and .
Step 6.4
Move the negative in front of the fraction.
Step 6.5
Cancel the common factor of .
Step 6.5.1
Factor out of .
Step 6.5.2
Factor out of .
Step 6.5.3
Cancel the common factor.
Step 6.5.4
Rewrite the expression.
Step 6.6
Combine and .
Step 6.7
Move the negative in front of the fraction.
Step 6.8
Cancel the common factor of .
Step 6.8.1
Factor out of .
Step 6.8.2
Factor out of .
Step 6.8.3
Cancel the common factor.
Step 6.8.4
Rewrite the expression.
Step 6.9
Combine and .