# What is the Invertible Matrix?

## Matrix

Let us first know what a Matrix is before discussing the invertible matrix. A matrix in Mathematics is defined as the arrangement of numbers in rows and columns in a rectangular way. For example, the matrix with 2 rows and 3 columns is referred to as two by three matrix or 2 3 matrices.

An mn matrix X is known as an invertible matrix if only there exist an mn Y matrix such that:

**XY = YX = Iₙ**

Here Iₙ is known as identity matrix. The matrix Y is known as the inverse matrix of X.

## Inverse Matrix Application

- Inverse matrix is widely used in the different fields throughout linear algebra including similar matrices, diagonalizable matrices, and also used in the linear transformation that includes matrices.
- An invertible matrix is often used to encrypt or decrypt message codes.
- An invertible matrix is also used to explore electronic circuits, quantum mechanics, and optics.

## How Does Inverse Matrix Works?

The process of determining matrix inverse is known as invertible matrices. It should be considered that all matrices are not invertible. For a matrix to be invertible, it should be multiplied by its inverse. For example, there is no such number exists that multiplies with the number 0 and obtains the value 1. There, the digit 0 has no multiplicative inverse. Also, the matrices that have a different number of rows and columns have no multiplicative inverse.

# Multiplication of Matrix

As we all know matrix is an arrangement of numbers:

Consider the matrix with two 2 rows and 3 columns.

Multiplication of a 2 3 matrix by a single number, for example, “3” is quite easy.

In this case, “3” is known as a scalar. Hence, this type of matrix is known as scalar multiplication.

## Multiplication of Matrix by Another Matrix

Multiplication of a matrix by another matrix, we need to do a dot product of both columns. In dot product, we multiply the rows of the first matrix with the column of the second matrix and then sum them up.

## Rules For Multiplication of Matrix

- The number of columns of the first matrix must be equal to the number of rows of the second matrix.
- And the result obtained after multiplying the two matrices will have the same number of rows as the first matrix and the same number of columns as the second matrix.

## Order For Multiplication of Matrix

In arithmetic, the commutative property of multiplication states that the product of two numbers remains the same regardless of their order. But this property does not hold true in the case of matrices. These properties can be understood in detail at Cuemath. When we change the order of multiplication in matrices, we get a different answer. Therefore: XY YX.

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