<< Chapter < Page | Chapter >> Page > |
What you just witnessed is no coincidence. This is the method that is often employed in finding the inverse of a matrix.
We list the steps, as follows:
Given the matrix below, find its inverse.
We write the augmented matrix as follows.
We reduce this matrix using the Gauss-Jordan method.
Multiplying the first row by –2 and adding it to the second row, we get
If we swap the second and third rows, we get
Divide the second row by –2. The result is
Let us do two operations here. 1) Add the second row to first, 2) Add -5 times the second row to the third. And we get
Multiplication of the third row by 2 results in
Multiply the third row by and add it to the second. Also, multiply the third row by and add it to the first. We get
Therefore, the inverse of matrix is
One should verify the result by multiplying the two matrices to see if the product does, indeed, equal the identity matrix.
Now that we know how to find the inverse of a matrix, we will use inverses to solve systems of equations. The method is analogous to solving a simple equation like the one below.
Solve the following equation .
To solve the above equation, we multiply both sides of the equation by the multiplicative inverse of which happens to be . We get
We use the above example as an analogy to show how linear systems of the form are solved.
To solve a linear system, we first write the system in the matrix equation , where is the coefficient matrix, the matrix of variables, and the matrix of constant terms. We then multiply both sides of this equation by the multiplicative inverse of the matrix .
Consider the following example.
Solve the following system
To solve the above equation, first we express the system as
where is the coefficient matrix, and is the matrix of constant terms. We get
To solve this system, we multiply both sides of the matrix equation by . Since the matrix is the same matrix whose inverse we found in [link] ,
Multiplying both sides by , we get
Therefore, , and .
Solve the following system
To solve the above equation, we write the system in the matrix form as follows:
To solve this system, we need inverse of . From [link] , we have
We multiply both sides of the matrix equation , by , we get
After multiplying the matrices, we get
Notification Switch
Would you like to follow the 'Applied finite mathematics' conversation and receive update notifications?