1. Solving Linear Systems of Equations - Inverse Matrix
Consider the following system of equations ...
Let the matrix A represent the coefficients ...
Let matrix B hold the constants ...
Finally, let matrix X represent the variables ...

2. Solving Linear Systems of Equations - Inverse Matrix
Now notice what the result is when we work out the following matrix equation ...
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3. Solving Linear Systems of Equations - Inverse Matrix
Thus, AX = B represents the system of equations. This matrix equation can be solved for X as follows ...
Recall that matrix multiplication is not commutative, so each side of the equation must be multiplied on the left by A-1
Matrix multiplication is associative.
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4. Solving Linear Systems of Equations - Inverse Matrix
Method of solution:
(1) Given a system of equations, form matrices
A, X, and B.
A Coefficients
X Variables (vertical matrix)
B Constants (vertical matrix)
(2) Find A-1.
(3) Find the solution by multiplying A-1 times B.
X = A-1 B
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5. Solving Linear Systems of Equations - Inverse Matrix
Example:
Use an inverse matrix to solve the
system at the right.
Using the methods of finding an inverse, A-1 is ...
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6. Solving Linear Systems of Equations - Inverse Matrix
Now find X ...
The solution is (2, -1), or
x = 2
y = -1
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7. Solving Linear Systems of Equations - Inverse Matrix
This same method can be used on any size system of equations as long as the coefficient matrix is square and the solution is unique.
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Solving Linear Systems of Equations - Inverse Matrix
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