In this explainer, we will learn how to solve a system of two linear equations using the inverse of the matrix of coefficients.
The translation between systems and matrix equations is straightforward. Consider the system
We can check that the values , , solve this system by verifying that it produces identities and and
We can also write the system as an equation between two matrices:
Then seeing the left matrix as a product of a matrix and a matrix separates the coefficients from the unknowns:
In other words, we have expressed our system as the matrix equation with the form where is a matrix, while and are both matrices. Here, is the (matrix of) unknowns and the right-hand side of the equation, and this is exactly like the equation which has form with numbers instead of matrices.
This is the simplest kind of linear equation and we know that provided the coefficient (here 2) is nonzero, we can solve this by multiplying the equation through by the multiplicative inverse of that coefficient: is the same as
We can use the same method here too. In the place of requiring that so that exists, we demand that have an inverse matrix so that , the identity matrix.
Now that we have we continue to solve for : is the same as which gives the same solution, this time assembled into a matrix.
Example 1: Expressing a Set of Simultaneous Equations as a Matrix Equation
Express the given set of simultaneous equations as a matrix equation.
The important thing is to be sure that the system of equations has the unknowns in the same order for each equation. Then the coefficient matrix is easy to read off the matrix equation:
Translating in the other direction is a similar exercise.
Example 2: Identifying a Set of Simultaneous Equations from a Matrix Equation
Write down the set of simultaneous equations that could be solved using the matrix equation
Writing the three components of the matrix equation, we get the following system on unknowns : which, of course, is the system of equations. As a list, it becomes
Generally, the bulk of the work is in inverting the matrix.
Example 3: Solving a Matrix Equation by Finding the Inverse
Given that find the values of , and .
Of course, the solution exists provided that the coefficient matrix above is invertible. This turns out to be the case, with such that
Therefore, the system is solved by