What is a Matrix Inverse?
In matrix algebra, division is not defined directly. Instead, we use the concept of a matrix inverse. The inverse of a matrix A (written as A-1) acts like the reciprocal of a number. Just as multiplying a number by its reciprocal yields 1 (e.g., 5 × 1/5 = 1), multiplying a matrix by its inverse yields the Identity matrix (I).
Only square matrices can have inverses, but not all square matrices do. A matrix that has an inverse is called invertible or non-singular. A matrix that does not have an inverse is called singular.
Finding the Inverse via Gauss-Jordan Elimination
The most robust method for computing the inverse of an n x n matrix by hand is the augmented matrix method. Here is the step-by-step process:
- Create an augmented matrix of size n x 2n by placing the original matrix A on the left and the n x n Identity matrix I on the right: [A | I].
- Perform Gauss-Jordan elimination on this augmented matrix to reduce the left side to its Reduced Row Echelon Form (RREF).
- If the matrix A is invertible, the left side will reduce completely to the Identity matrix I. When this happens, the right side will automatically transform into the inverse matrix A-1: [I | A-1].
- If the left side cannot be reduced to the Identity matrix (e.g., if you get a row of all zeros on the left side), the matrix is singular, and no inverse exists.
Algebraic Properties of Inverses
- (A-1)-1 = A (the inverse of an inverse is the original matrix).
- (AB)-1 = B-1A-1 (the inverse of a product is the product of the inverses in reverse order).
- (AT)-1 = (A-1)T (the inverse of the transpose is the transpose of the inverse).