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Linear Algebra Examples
Step 1
Step 1.1
The inverse of a matrix can be found using the formula where is the determinant.
Step 1.2
Find the determinant.
Step 1.2.1
The determinant of a matrix can be found using the formula .
Step 1.2.2
Simplify the determinant.
Step 1.2.2.1
Simplify each term.
Step 1.2.2.1.1
Cancel the common factor of .
Step 1.2.2.1.1.1
Factor out of .
Step 1.2.2.1.1.2
Cancel the common factor.
Step 1.2.2.1.1.3
Rewrite the expression.
Step 1.2.2.1.2
Cancel the common factor of .
Step 1.2.2.1.2.1
Move the leading negative in into the numerator.
Step 1.2.2.1.2.2
Factor out of .
Step 1.2.2.1.2.3
Cancel the common factor.
Step 1.2.2.1.2.4
Rewrite the expression.
Step 1.2.2.1.3
Multiply by .
Step 1.2.2.2
Subtract from .
Step 1.3
Since the determinant is non-zero, the inverse exists.
Step 1.4
Substitute the known values into the formula for the inverse.
Step 1.5
Move the negative in front of the fraction.
Step 1.6
Multiply by each element of the matrix.
Step 1.7
Simplify each element in the matrix.
Step 1.7.1
Multiply .
Step 1.7.1.1
Multiply by .
Step 1.7.1.2
Multiply by .
Step 1.7.2
Cancel the common factor of .
Step 1.7.2.1
Move the leading negative in into the numerator.
Step 1.7.2.2
Factor out of .
Step 1.7.2.3
Cancel the common factor.
Step 1.7.2.4
Rewrite the expression.
Step 1.7.3
Multiply by .
Step 1.7.4
Multiply .
Step 1.7.4.1
Multiply by .
Step 1.7.4.2
Multiply by .
Step 1.7.4.3
Multiply by .
Step 1.7.4.4
Multiply by .
Step 1.7.5
Cancel the common factor of .
Step 1.7.5.1
Move the leading negative in into the numerator.
Step 1.7.5.2
Factor out of .
Step 1.7.5.3
Cancel the common factor.
Step 1.7.5.4
Rewrite the expression.
Step 1.7.6
Multiply by .
Step 2
Multiply both sides by the inverse of .
Step 3
Step 3.1
Multiply .
Step 3.1.1
Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is and the second matrix is .
Step 3.1.2
Multiply each row in the first matrix by each column in the second matrix.
Step 3.1.3
Simplify each element of the matrix by multiplying out all the expressions.
Step 3.2
Multiplying the identity matrix by any matrix is the matrix itself.
Step 3.3
Multiply .
Step 3.3.1
Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is and the second matrix is .
Step 3.3.2
Multiply each row in the first matrix by each column in the second matrix.
Step 3.3.3
Simplify each element of the matrix by multiplying out all the expressions.