How to calculate the inverse of a 2x2 matrix

Suppose we have a matrix M = [{a b} {c d}]

The inverse of a matrix M is any matrix, that when multiplied by M, gives the unit matrix (I) - in this case: [{1 0} {0 1}]

First, we have to determine if the matrix can be inverted or not. A matrix can only be inverted if it is square, and if the determinant is not zero (the determinant of a matrix is analogous to a single numeric value, representing the "size" of a matrix. the inverse of 0 makes no sense, as 1/0 is undefined)

The determinant of M, det(M), is calculated as follows:

det(M) = ad-bc.

Now, to calculate the inverse of M. In general, there are many methods for calculating inverse matrices, and these methods get progressively more complicated the larger the matrix. However, for a 2x2 matrix, there exists a simple method:

inverse of M = (1/det(M))[{d -b} {-c a}]

The top left and bottom right values are swapped, and the top right and bottom left values are multiplied by -1. Then every value of the matrix is divided by the determinant of the original matrix.

It is important to note that this method only works for 2x2 matrices - trying it with any other type of matrix would yield false results, if any.