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
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
only one solution
the solution is the point of intersection between the 2 lines
there is no intersection points between the lines
infinitly many solutions
the 2 lines overlap
¨ General system of n equations and m unknowns.
¨ Any system of equations can be written as an augmented matrix. A matrix is just a rectangular array of numbers.
¨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.
Solving the system
¨To solve a system using a matrix
- Make 0's in the three lower left positions, that is, in the positions below the upper left to lower-right diagonal
- Make 1's on the upper left to lower right diagonal.