Linear Algebra Examples

Solve the Matrix Equation [[-4,-5,5],[1,4,-2],[6,1,-6]][[x],[y],[z]]=[[-11],[3],[17]]
Step 1
Multiply .
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Step 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 1.2
Multiply each row in the first matrix by each column in the second matrix.
Step 1.3
Simplify each element of the matrix by multiplying out all the expressions.
Step 2
Write as a linear system of equations.
Step 3
Solve the system of equations.
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Step 3.1
Move all terms not containing to the right side of the equation.
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Step 3.1.1
Subtract from both sides of the equation.
Step 3.1.2
Add to both sides of the equation.
Step 3.2
Replace all occurrences of with in each equation.
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Step 3.2.1
Replace all occurrences of in with .
Step 3.2.2
Simplify the left side.
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Step 3.2.2.1
Simplify .
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Step 3.2.2.1.1
Simplify each term.
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Step 3.2.2.1.1.1
Apply the distributive property.
Step 3.2.2.1.1.2
Simplify.
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Step 3.2.2.1.1.2.1
Multiply by .
Step 3.2.2.1.1.2.2
Multiply by .
Step 3.2.2.1.1.2.3
Multiply by .
Step 3.2.2.1.2
Simplify by adding terms.
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Step 3.2.2.1.2.1
Subtract from .
Step 3.2.2.1.2.2
Add and .
Step 3.2.3
Replace all occurrences of in with .
Step 3.2.4
Simplify the left side.
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Step 3.2.4.1
Simplify .
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Step 3.2.4.1.1
Simplify each term.
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Step 3.2.4.1.1.1
Apply the distributive property.
Step 3.2.4.1.1.2
Simplify.
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Step 3.2.4.1.1.2.1
Multiply by .
Step 3.2.4.1.1.2.2
Multiply by .
Step 3.2.4.1.1.2.3
Multiply by .
Step 3.2.4.1.2
Simplify by adding terms.
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Step 3.2.4.1.2.1
Add and .
Step 3.2.4.1.2.2
Subtract from .
Step 3.3
Solve for in .
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Step 3.3.1
Move all terms not containing to the right side of the equation.
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Step 3.3.1.1
Subtract from both sides of the equation.
Step 3.3.1.2
Subtract from both sides of the equation.
Step 3.3.1.3
Subtract from .
Step 3.3.2
Divide each term in by and simplify.
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Step 3.3.2.1
Divide each term in by .
Step 3.3.2.2
Simplify the left side.
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Step 3.3.2.2.1
Cancel the common factor of .
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Step 3.3.2.2.1.1
Cancel the common factor.
Step 3.3.2.2.1.2
Divide by .
Step 3.3.2.3
Simplify the right side.
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Step 3.3.2.3.1
Simplify each term.
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Step 3.3.2.3.1.1
Dividing two negative values results in a positive value.
Step 3.3.2.3.1.2
Dividing two negative values results in a positive value.
Step 3.4
Replace all occurrences of with in each equation.
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Step 3.4.1
Replace all occurrences of in with .
Step 3.4.2
Simplify the left side.
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Step 3.4.2.1
Simplify .
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Step 3.4.2.1.1
Simplify each term.
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Step 3.4.2.1.1.1
Apply the distributive property.
Step 3.4.2.1.1.2
Combine and .
Step 3.4.2.1.1.3
Multiply .
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Step 3.4.2.1.1.3.1
Combine and .
Step 3.4.2.1.1.3.2
Multiply by .
Step 3.4.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 3.4.2.1.3
Combine and .
Step 3.4.2.1.4
Combine the numerators over the common denominator.
Step 3.4.2.1.5
Simplify the numerator.
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Step 3.4.2.1.5.1
Multiply by .
Step 3.4.2.1.5.2
Add and .
Step 3.4.2.1.6
Move the negative in front of the fraction.
Step 3.4.2.1.7
To write as a fraction with a common denominator, multiply by .
Step 3.4.2.1.8
Combine and .
Step 3.4.2.1.9
Combine the numerators over the common denominator.
Step 3.4.2.1.10
Combine the numerators over the common denominator.
Step 3.4.2.1.11
Multiply by .
Step 3.4.2.1.12
Subtract from .
Step 3.4.2.1.13
Rewrite as .
Step 3.4.2.1.14
Factor out of .
Step 3.4.2.1.15
Factor out of .
Step 3.4.2.1.16
Move the negative in front of the fraction.
Step 3.4.3
Replace all occurrences of in with .
Step 3.4.4
Simplify the right side.
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Step 3.4.4.1
Simplify .
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Step 3.4.4.1.1
Simplify each term.
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Step 3.4.4.1.1.1
Apply the distributive property.
Step 3.4.4.1.1.2
Combine and .
Step 3.4.4.1.1.3
Multiply .
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Step 3.4.4.1.1.3.1
Combine and .
Step 3.4.4.1.1.3.2
Multiply by .
Step 3.4.4.1.1.4
Simplify each term.
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Step 3.4.4.1.1.4.1
Move the negative in front of the fraction.
Step 3.4.4.1.1.4.2
Move the negative in front of the fraction.
Step 3.4.4.1.2
To write as a fraction with a common denominator, multiply by .
Step 3.4.4.1.3
Combine and .
Step 3.4.4.1.4
Combine the numerators over the common denominator.
Step 3.4.4.1.5
Simplify the numerator.
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Step 3.4.4.1.5.1
Multiply by .
Step 3.4.4.1.5.2
Subtract from .
Step 3.4.4.1.6
To write as a fraction with a common denominator, multiply by .
Step 3.4.4.1.7
Combine and .
Step 3.4.4.1.8
Combine the numerators over the common denominator.
Step 3.4.4.1.9
Combine the numerators over the common denominator.
Step 3.4.4.1.10
Multiply by .
Step 3.4.4.1.11
Add and .
Step 3.5
Solve for in .
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Step 3.5.1
Multiply both sides of the equation by .
Step 3.5.2
Simplify both sides of the equation.
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Step 3.5.2.1
Simplify the left side.
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Step 3.5.2.1.1
Simplify .
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Step 3.5.2.1.1.1
Cancel the common factor of .
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Step 3.5.2.1.1.1.1
Move the leading negative in into the numerator.
Step 3.5.2.1.1.1.2
Factor out of .
Step 3.5.2.1.1.1.3
Cancel the common factor.
Step 3.5.2.1.1.1.4
Rewrite the expression.
Step 3.5.2.1.1.2
Multiply.
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Step 3.5.2.1.1.2.1
Multiply by .
Step 3.5.2.1.1.2.2
Multiply by .
Step 3.5.2.2
Simplify the right side.
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Step 3.5.2.2.1
Multiply by .
Step 3.5.3
Move all terms not containing to the right side of the equation.
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Step 3.5.3.1
Subtract from both sides of the equation.
Step 3.5.3.2
Subtract from .
Step 3.5.4
Divide each term in by and simplify.
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Step 3.5.4.1
Divide each term in by .
Step 3.5.4.2
Simplify the left side.
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Step 3.5.4.2.1
Cancel the common factor of .
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Step 3.5.4.2.1.1
Cancel the common factor.
Step 3.5.4.2.1.2
Divide by .
Step 3.5.4.3
Simplify the right side.
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Step 3.5.4.3.1
Divide by .
Step 3.6
Replace all occurrences of with in each equation.
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Step 3.6.1
Replace all occurrences of in with .
Step 3.6.2
Simplify the right side.
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Step 3.6.2.1
Simplify .
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Step 3.6.2.1.1
Simplify the numerator.
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Step 3.6.2.1.1.1
Multiply by .
Step 3.6.2.1.1.2
Subtract from .
Step 3.6.2.1.2
Divide by .
Step 3.6.3
Replace all occurrences of in with .
Step 3.6.4
Simplify the right side.
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Step 3.6.4.1
Simplify .
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Step 3.6.4.1.1
Combine the numerators over the common denominator.
Step 3.6.4.1.2
Simplify the expression.
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Step 3.6.4.1.2.1
Multiply by .
Step 3.6.4.1.2.2
Subtract from .
Step 3.6.4.1.2.3
Divide by .
Step 3.7
List all of the solutions.