## Introduction

Today we will discuss the working of Matrix Calculator. Matrix math is actually a collection of elements, arranged in rows and columns and the combination of linear equation. In matrix math,

the numbers 1, 2, 3, 4,…..n

are called the entries or elements of the matrix. Matrix represented by the following symbols **()**,**[]**,**|| ||**,e.g.

$$\left[\begin{matrix} 2 & 1 & 3 \\ 4 & 2 & 1 \\ 3 & 3 & 2 \\ \end{matrix}\right] 3*3 $$

There are many types of a matrix for example; row matrix, column matrix, square matrix, null matrix, identity matrix, diagonal matrix, scalar matrix, triangular matrix, transpose matrix, symmetric matrix, skew matrix, equal matrix, algebraic matrix. Now we will know how matrix calculator performs its function. No doubt, matrices have various applications not only in different branches of mathematics but also in physics, economics, engineering, statistics and economics. So, matrix solver has it’s actual benefits nowadays.

## How Matrix Calculator Multiply Matrices Equations

Let us start to know the working of matrix multiplication calculator. First I will let you know about the sizes of each matrix because not every matrix can be multiplied by another matrix. So, let’s take a look at the size of this first matrix which has 2 rows and 3 columns and the second one matrix has 3 rows and 3 columns. So the first matrix is a two by three matrix. If we take a look at the second matrix, it is three by three matrix. So, in order to multiply two matrices together the columns in the first matrices must equal the rows in the second matrix.

$$\left[\begin{matrix} 2 & 3 & 1 \\ 2 & -7 & 4 \\ \end{matrix}\right].\left[\begin{matrix} 3 & 4 & 5 \\ 1 & 1 & 4 \\ 2 & 1 & 4 \\ \end{matrix}\right]$$

These two inside numbers must be the same. If these two numbers are not the same, you can’t multiply the two matrices. So, make sure about these numbers are the same, before multiply matrices. The outside numbers give you the size of the new matrix after you have done multiplication. Here outside numbers are 2 and 3, so the new matrix size will be 2 by 3 matrix.

Now let us take start, what actually we do when we multiply two matrices. We will multiply the rows of the first matrix by the columns of the second one

$$\left[\begin{matrix} 2 & 3 & 1 \\ 2 & -7 & 4 \\ \end{matrix}\right].\left[\begin{matrix} 3 & 4 & 5 \\ 1 & 1 & 4 \\ 2 & 1 & 4 \\ \end{matrix}\right]$$

We will multiply 2 of the first matrix to 3 of the 2nd matrix and so on. I highlighted them with the same color to make it easy to understand.

Multiply the next columns with the same row.

After that, apply multiplication to the 2nd row of first matrix with all columns of the 2nd matrix.

**Finally,**

$$=\left[\begin{matrix}11& 12& 26\\7&5&-2 \end{matrix}\right]$$

## Inverse Matrix Calculator

In an A is a n*n matrix, I also a n*n matrix, in this situation the n*n matrix B. It will be denoted as B = A^{-1} and it will be said as an inverse matrix ( AA^{-1} = A^{-1} A = I). Inverse matrix is denoted by A^{-1}.

To obtain inverse matrix A for which A^{-1} exists, the inverse matrix calculator do the following steps. First of all, form an augmented [ A/I] matrix in which I is an n*n identity matrix. Then, to get a matrix [I/B], perform row transformations on [ A / I]. Actually here, matrix B is A^{-1}. At last step, verify it by showing that these matrices are equal to each other e.g. B.A = A.B = I.

One thing must be noted that every matrix does not have inverses but only square matrices. It is not necessary that every matrix has an inverse. And it’s the special property of an inverse matrix that it is unique but if it exists. Any matrix has no more than one matrix.