Representing Systems of Linear Equations using Matrices
A system of linear equations can be represented in matrix form using a coefficient matrix, a variable matrix, and a constant matrix.
Consider the system,
The coefficient matrix can be formed by aligning the coefficients of the variables of each equation in a row. Make sure that each equation is written in standard form with the constant term on right.
Then, the coefficient matrix for the above system is
The variables we have are and . So we can write the variable matrix as .
On the right side of the equality we have the constant terms of the equations, and . The two numbers in that order correspond to the first and second equations, and therefore take the places at the first and the second rows in the constant matrix. So, the matrix becomes .
Now, the system can be represented as .
Using matrix multiplication you can see that the matrix representation is equivalent to the system of equations.
That is, .
Equating the corresponding entries of the two matrices we get:
Now let us understand what this representation means.
If you consider this as a function of the vector , it can be defined as
Then, by solving the system what we are finding a vector for which .
This representation can make calculations easier because, if we can find the inverse of the coefficient matrix, the input vector can be calculated by multiplying both sides by the inverse matrix.
In a similar way, for a system of three equations in three variables,
The matrix representation would be
We can generalize the result to variables.