A matrix is a rectangular arrangement of elements, where the number of rows and columns defines its order (m x n). Basic arithmetic operations such as addition, subtraction, and multiplication can be performed on matrices. But, there is another type of operation known as
elementary operations
which are performed on rows and columns of matrices.
The inverse of a matrix A, denoted as A-1, is defined such that when multiplied with A, results in the identity matrix (I). That is, A.A-1 = I.
We will now explore how to find the inverse of a matrix using elementary operations.
The Basics of Elementary Operations or Transformations
Elementary operations or transformations include six operations that can be performed on a matrix. These operations are applicable only to
square matrices
.
The elementary operations include:
Swapping any two rows or two columns
Multiplying the elements of any row or column by a non-zero constant
Adding or subtracting multiples of one row to another
How to Find the Inverse of a Matrix Using Elementary Operations
Consider a matrix equation X = AB, where X, A and B are matrices of the same order. To find the inverse, we apply elementary row operations on X and A (on the right side of the equation), and elementary column operations on X and B simultaneously.
The end goal is to apply a sequence of elementary row operations on A = IA or elementary column operations on A = AI until we end up with I = BA or I = AB respectively. This gives us the inverse of matrix A.
Note:
If A and B are two square matrices such that AB = BA = I, then B is the inverse matrix of A and vice versa.