The inverse of a matrix
Why inverse matrices?
The inverse of a matrix is also called the reciprocal of the matrix. Since matrices cannot be divided, inverse matrices are used as the reciprocal operation of multiplication of matrices.
Notation If A is a square matrix, then its inverse is also a square matrix and it is denoted by A^{−1}.
AA^{−1} = I, where I is the identity matrix.
For example, in the equation AX = B, where A, X and B are matrices, X = A^{−1}B.
Invertible matrices must be square matrices (the number of rows equals the number of columns).
Here are some square matrices:
(2 × 2)  (3 × 3) 
(4 × 4) 

The determinant of an invertible matrix must have a nonzero value (determinant ≠ 0)
If the determinant of a matrix is equal to zero, then the matrix is called a singular matrix.
Notation If A is a square matrix, then its inverse is also a square matrix and it is denoted by A^{−1}.
The formula required to calculate the inverse of a (2 × 2) matrix is shown below.
If  , then the inverse matrix,  . 
Example
Determine the inverse matrix of the (2 × 2) square matrix, M, given below.
Step 1. Calculate the determinant of matrix M.
det(M)  = (−2) × 3 − (−1) × 5 
= −6 + 5  
= −1 
Step 2. Swap the two elements on the leading diagonal.
Step 3. Change the signs of the elements on the second diagonal.
Step 4. Write the inverse matrix using the formula given.
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