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a 3x +5y - 2z from 8x +11y +5zb׳- 2x²+ 12x-5 from 5x²-7-3x³+2x
to the science of mathematics and physics span fields such as algebra, number theory, analysis, differential geometry, astronomy, and optics, among others. His discoveries regarding matrix theory changed the way mathematicians have worked for the last two centuries.
In this section, we will learn Gaussian elimination for solving systems, using matrices.
Writing the Augmented Matrix of a System of Equations
A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix.
For example, consider the following 2 × 2system of equations.
3x + 4y = 7
4x − 2y = 5
We can write this system as an augmented matrix (please note that a more common formatting for augmented matrices has a solid vertical line running through the matrix, rather than a line on each row):
[
3
4
|
7
4
−
2
|
5
]
We can also write a matrix containing just the coefficients. This is called the
coefficient matrix.
[
3
4
4
−
2
]
A three-by-three system of equations such as
3x − y − z = 0
x + y = 5
2x − 3z = 2
has a coefficient matrix
⎡
⎢
⎣
3
−
1
−
1
1
1
0
2
0
−
3
⎤
⎥
⎦
and is represented by the augmented matrix
⎡
⎢
⎣
3
−
1
−
1
∣
0
1
1
0
∣
5
2
0
−
3
∣
2
⎤
⎥
⎦
Notice that the matrix is written so that the variables line up in their own columns:
x-terms go in the first column, y-terms in the second column, and z-terms in the third column. It is very important that each equation is written ax + by + cz = d in standard form so that the variables line up. When there is a missing variable term in an equation, the coefficient is 0.