Inverse of Matrix Calculator

Lets first understand, what is the inverse of a matrix and how we can find the inverse of a matrix?

Definition: A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.

A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix (read as an m by n matrix).

We denote matrices by capital letters. Some examples of matrices:

\[ A= \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}, B=\begin{bmatrix} g & h \\ i & j \\ k & l \end{bmatrix} \] \[ C= \begin{bmatrix} m & n & o \\ p & q & r \\ s & t & u \end{bmatrix} \]

Square matrix

A matrix in which the number of rows are equal to the number of columns, is said to be a square matrix.

\[ P= \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]

Identity matrix

A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix.

\[ I_{1}=\begin{bmatrix} 1 \end{bmatrix},I_{2}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \] \[ I_{3}= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

Ones matrix

A matrix is said to be ones matrix if all its elements are one.

\[ A= \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix} \]

Invertible matrices

Definition: If A is a square matrix of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is called the inverse matrix of A and it is denoted by A^-1. In that case A is said to be invertible.

\[ A=\begin{bmatrix} 4 & 3 \\ 2 & 1 \end{bmatrix}, B=\begin{bmatrix} -0.5 & 1.5 \\ 1 & -2 \end{bmatrix} \] \[ AB= BA= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} =I\]

Note:

1. A rectangular matrix does not possess inverse matrix, since for products BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order.

2. If B is the inverse of A, then A is also the inverse of B.

For calculating the inverse of a matrix using above matrix inverse calculator, you have to first set the order of matrix, then enter the matrix and press confirm button, after that choose the operation, then you will get the resultant matrix in the Result section.