Inverse of a Matrix

The first condition for the inverse of a matrix to exist is that it must be a square matrix. This doesn’t mean that the inverse of every square matrix exists. The inverse of a square matrix exists if and only the square matrix is non-singular. A matrix is said to be singular if its determinant is equal to zero, on the other hand, a matrix is said to be non-singular if its determinant is not zero.

Thus, in general, we can say that the inverse of every non-singular square matrix exists. The inverse of a non-singular square matrix is a matrix which when multiplied with the given matrix results in an identity matrix.

Inverse of a matrix

\[Let\:A=\begin{bmatrix}
1&5&2\\
6&7&8\\
2&9&4\\
\end{bmatrix}\]

and determinant of A = -4

Since the determinant of a square matrix A is -4 which means that A is non-singular. So, A-1 exists which is shown below;

A square matrix is invertible if the matrix is non-singular. When a square matrix is non-singular, then its inverse by the adjoint method is calculated with the help of the following formula;

Inverse of a Matrix

Where ∣A∣ is the determinant of ‘A’ matrix and ‘adj A’ is the adjoint of the matrix A.

Let A is a non-singular square matrix of order “n” and AB = I, then matrix B of the same order ‘n’ is said to be the inverse of matrix A where ‘I’ is an identity or unit matrix. Also, AB is the product of matrices A & B.

The formula given above is used to compute the inverse of a matrix. If you want to compute the inverse of a matrix by using this formula then you must know about the minor, cofactor and adjoint of the matrix. There are various other ways by which we can calculate the inverse of a matrix. Those methods will be discussed in separate articles on this blog. You can also calculate the inverse of any matrix online.

if you have any questions regarding this article you can leave your questions below in the comment section.

Leave a Comment