Inverse matrix method. Here it’s about the inverse matrix method to solve a system of equations.
Inverse matrix method
There are several ways to solve a set of equations in matrix algebra like Gaussian elimination method, row transformation method, triangular decomposition method and so on. The inverse matrix method is also one of that.
Here I’ll explain how to use the inverse matrix method to solve a set of equations.
Suppose I have a set of equations like
Now I have to solve these equations using the inverse matrix method.
First of all, I’ll write the set of equations in matrix form.
Thus it will be
So I can say the system of equations is in the form of .
Here is the coefficient matrix, is the variable matrix and is the constant matrix.
Thus the value of the variable matrix will be .
Now is the inverse of the matrix .
So it will be a straightforward one to handle.
But please also note that this method is only good up to three variables. If the number of variables is higher than three, this particular method is no good trying.
Now I’ll solve an example of this method.
Example of the inverse matrix method
Disclaimer: This is not my own example. I have chosen it from some book. I have also given the due reference at the end of the post.
Here is an example.
According to Stroud and Booth (2011)* “If where
determine and hence solve the set of equations.”
Here I know both the coefficient matrix and the constant matrix .
Also, I have to find out the value of the variable matrix .
First of all, I’ll determine the value of the inverse of the matrix .
So, I’ll start with the determinant of the matrix .
Here the given matrix is
Thus the determinant of the matrix will be
Now I will evaluate the determinant. This gives
Next, I will find out the matrix with the cofactors.
First of all, I’ll find out the cofactors in row 1. These are and .
So I get
Now I’ll find out the cofactors in row 2. These are and .
So I get
I’ll find out the cofactors in row 3. These are and .
So I get
Thus the matrix will be
As a next step, I’ll get the transpose of the matrix . And then I’ll determine the inverse of the matrix .
Now to get the transpose of the matrix , I’ll interchange between its rows and columns.
Thus the transposed matrix will be
Therefore the inverse of the matrix will be
So this gives
Now I’ll get the value of the variable matrix .
As I have already said before, , this gives .
Thus the value of is
Dear friends, this is the end of my today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!