Linear algebra

Solving systems by using matrices

Definition of a linear equation

¨For an equation to be a linear equation.

¨1) Unknowns are numerators,

¨2) Unknowns are only to the first power

¨3) There are no products or quotients of unknowns.

¨The most general linear equation is,

¨where there are n unknowns with known coefficients

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A system of n equations and m unknowns

¨Theorem 1 Given a system of n equations and m unknowns there will be one of three possibilities for solutions to the system.

¨There will be no solution (called inconsistent)

¨There will be exactly one solution (consistent)

¨There will be infinitely many solutions (consistent)

For a system of 2 equations with 2 unknowns

Matrix

¨ General system of n equations and m unknowns.

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¨ Any system of equations can be written as an augmented matrix. A matrix is just a rectangular array of numbers.

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¨

¨Each row of the augmented matrix consists of the coefficients and constant on the right of the equal sign from a given equation in the system. The first row is for the first equation, the second row is for the second equation etc. Likewise each of the first m columns of the matrix consists of the coefficients from the unknowns. The first column contains the coefficients of , the second column contains the coefficients of , etc. The final column (the m+1st column) contains all the constants on the right of the equal sign.

¨

¨And we call this the coefficient matrix for the system.

¨

¨

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Matrix operations

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Solving the system


¨To solve a system using a matrix



  1. Make 0's in the three lower left positions, that is, in the positions below the upper left to lower-right diagonal
  2. Make 1's on the upper left to lower right diagonal.


When the number of equations and unknown is big, solving the system using a matrix is much easier and faster than the any other way since it can directly find the solution or even show if the system doesn’t have a solution when it gives nonsense