The Inverse of a Matrix

Definition

The inverse of a square matrix  of order (n x n) is denoted by ^-1. The matrix is invertable if ^-1 exists such that; ^-1== ^-1,  where I  is an Identity matrix of order (n x n)

General Theorem

If A is an (n x n) matrix, then A is invertable if the equation A x=b has a unique solution;

x=^-1b

The unique inverse of an invertable matrix  is denoted by^-1.

Evaluating the Inverse

A good method for finding the inverse of an (n x n) matrix is to form the (n x 2n)-matrix (|), where   is the n x n matrix and  is an n x n unit matrix.

We must then reduce ( | ) to row-echelon form by successive applications of elementary row operations, which consist of multipling rows by constants and add multiples of one row to another row.

Matrix A is invertable only if ( |) reduces to;

(|^-1)

Then^-1 is the inverse of matrix .

Example

We then apply elementary row operations:

By subtracting 3 x Row 1 from Row 2 we obtain:

By dividing Row 2 by -3 we obtain

By subtracting Row 2 from Row 1 we obtain:

The last result is in the form:  therefore:

Properties of the Inverse:

•  The inverse of a lower triangular matrix is also a lower triangular matrix, and the inverse of an upper triangular matrix is likewise an upper tiangular matrix. Consider a lower triangular matrix in the following example:

= and ^-1=

The opposite is also true:

= and ^-1= 

Note: The inverse of a diagonal matrix is also a diagonal matrix.

• The inverse of the product of two matrices is equal to the product of their inverses: ()^-1=^-1-1
• The inverse the product of a scalar (k) and a matrix is equal to the porduct of the reciprocal of the constant and the inverse of the matrix: (k)^-1=(1/k)^-1
• The inverse of the transpose of a matrix is equal to the transpose of the inverse of that matrix: (^T)^-1=(^-1)^T
• There is a unique solution for = if is invertable, the solution is =^-1