A matrix is a rectangular arrangement of numbers into rows and columns.
Matrices can be used to solve systems of equations. But first, we must learn how to represent systems with matrices.
Representing a linear system with matrices
A system of equations can be represented by an augmented matrix.
In an augmented matrix, each row represents one equation in the system and each column represents a variable or the constant terms.
In this way, we can see that augmented matrices are a shorthand way of writing systems of equations. The organization of the numbers into the matrix makes it unnecessary to write various symbols like xx, yy, and =equals , yet all of the information is still there!
Now that we have the basics, let's take a look at a slightly more complicated example.
Write the following system of equations as an augmented matrix.
To make things easier, let's rewrite the system to show each of the coefficients clearly. If a variable term is not written in an equation, it means that the coefficient is 00.
This corresponds to the following augmented matrix.
Again, notice how each column corresponds to a variable (xstart color greenD, x, end color greenD, ystart color purpleC, y, end color purpleC, zstart color goldD, z, end color goldD) or the constantsstart color blueD, c, o, n, s, t, a, n, t, s, end color blueD. Also notice that the numbers in each row correspond to the coefficients in the same equation.
In general, before converting a system into an augmented matrix, be sure that the variables appear in the same order in each equation, and that the constant terms are isolated on one side.
This problem differs in that each equation in the system is not currently in standard form. Let's start by writing each equation in standard form. This will help to align the variables and the constants.
Writing the first equation in standard form gives:
Writing the second equation in standard form gives:
So the system now looks like:
This corresponds to the following augmented matrix, or matrix CC.
Representing linear systems of equations with augmented matrices