Linear Algebra Examples

Solve Using an Inverse Matrix 1/2x+2/3y=-13/2 , 3x-y=6
,
Step 1
Find the from the system of equations.
Step 2
Find the inverse of the coefficient matrix.
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Step 2.1
The inverse of a matrix can be found using the formula where is the determinant.
Step 2.2
Find the determinant.
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Step 2.2.1
The determinant of a matrix can be found using the formula .
Step 2.2.2
Simplify the determinant.
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Step 2.2.2.1
Simplify each term.
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Step 2.2.2.1.1
Combine and .
Step 2.2.2.1.2
Move the negative in front of the fraction.
Step 2.2.2.1.3
Cancel the common factor of .
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Step 2.2.2.1.3.1
Factor out of .
Step 2.2.2.1.3.2
Cancel the common factor.
Step 2.2.2.1.3.3
Rewrite the expression.
Step 2.2.2.1.4
Multiply by .
Step 2.2.2.2
To write as a fraction with a common denominator, multiply by .
Step 2.2.2.3
Combine and .
Step 2.2.2.4
Combine the numerators over the common denominator.
Step 2.2.2.5
Simplify the numerator.
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Step 2.2.2.5.1
Multiply by .
Step 2.2.2.5.2
Subtract from .
Step 2.2.2.6
Move the negative in front of the fraction.
Step 2.3
Since the determinant is non-zero, the inverse exists.
Step 2.4
Substitute the known values into the formula for the inverse.
Step 2.5
Cancel the common factor of and .
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Step 2.5.1
Rewrite as .
Step 2.5.2
Move the negative in front of the fraction.
Step 2.6
Multiply the numerator by the reciprocal of the denominator.
Step 2.7
Multiply by .
Step 2.8
Multiply by each element of the matrix.
Step 2.9
Simplify each element in the matrix.
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Step 2.9.1
Multiply .
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Step 2.9.1.1
Multiply by .
Step 2.9.1.2
Multiply by .
Step 2.9.2
Multiply .
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Step 2.9.2.1
Multiply by .
Step 2.9.2.2
Multiply by .
Step 2.9.2.3
Multiply by .
Step 2.9.2.4
Multiply by .
Step 2.9.2.5
Multiply by .
Step 2.9.3
Multiply .
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Step 2.9.3.1
Multiply by .
Step 2.9.3.2
Combine and .
Step 2.9.3.3
Multiply by .
Step 2.9.4
Cancel the common factor of .
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Step 2.9.4.1
Move the leading negative in into the numerator.
Step 2.9.4.2
Factor out of .
Step 2.9.4.3
Cancel the common factor.
Step 2.9.4.4
Rewrite the expression.
Step 2.9.5
Move the negative in front of the fraction.
Step 3
Left multiply both sides of the matrix equation by the inverse matrix.
Step 4
Any matrix multiplied by its inverse is equal to all the time. .
Step 5
Multiply .
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Step 5.1
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 5.2
Multiply each row in the first matrix by each column in the second matrix.
Step 5.3
Simplify each element of the matrix by multiplying out all the expressions.
Step 6
Simplify the left and right side.
Step 7
Find the solution.