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Systems of equations with Matrices
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Solving and interpreting systems of linear equations through the lens of matrices.
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Writing a system in matrix form
AHL AI 1.14
A system of linear equations can be written in matrix form Ax=b. For a system of 3 equations with 3 unknowns, we let x=⎝⎛xyz⎠⎞, and then A is a matrix whose rows are the equations of the system, and columns the coefficients of each variable. The vector b represents the constants on the RHS of the equations
⎩⎪⎨⎪⎧x−z=43x+2y+z=2x−y−2z=3
can be written as
⎝⎛13102−1−11−2⎠⎞⎝⎛xyz⎠⎞=⎝⎛433⎠⎞
Solving a system using the inverse
AHL AI 1.14
Once a system of equations is written in the form Ax=b, so long as A−1 exists we can find the solution by left-multiplying both sides by the inverse:
x=A−1b
This solution is unique.
Interpreting the discriminant for a system of equations
AHL AI 1.14
When detA=0, A−1 does not exist. As such, a unique solution does not exist. Instead, the system has either no solutions, or it has infinitely many.
For systems of 2 or 3 equations, this can be interpreted as the lines / planes being parallel, and either lying on top of each other or having no intersections.
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