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Matrix Multiplication Calculator

Matrix Multiplication Calculator

Set matrices’ orders and write down their entities to find the product (if possible) up to 10*10 order through this matrix multiplication calculator.

Matrix A Dimension:


Matrix B Dimension:


Matrix A

Matrix B


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Allow this matrix multiplication calculator to find the product of two matrices that either contain complex numbers or not in seconds.

Here we will be discussing terms and conditions for matrix multiplication online. Moreover, we will see how to multiply matrices instantly with the help of this free matrix product calculator. So for a proper understanding of the whole scenario, keep yourself focused.

Let’s begin with a basic definition.

What Is A Matrix?

In the context of mathematics:
“A rectangular array or a formation of collection of real numbers, say 1 2 3 & 4 6 7, and then enclosed by the bracket [ ] is said to form a matrix”

For Example:

Let us represent all the numbers mentioned above in matrix form below:

$$ \begin{bmatrix} 1 & 2 & 3 \\ 4 & 6 & 7 \\\end{bmatrix} $$

Similarly we have some other matrices as below:

$$ \begin{bmatrix}10 & 10 \\ 8 & 8 \\\end{bmatrix} \hspace{0.25in} \begin{bmatrix} 6 \\ 3 \\\end{bmatrix} \hspace{0.25in} \begin{bmatrix} 2 \\\end{bmatrix} $$


Suppose we have two matrices as \(M_{1}\) and \(M_{2}\). Now if we multiply them, we will get a new matrix that is \(M_{3}\). The matrix multiplication is all about the product and addition of the elements of both matrices \(M_{1}\) and \(M_{2}\). All this generalization is as follows:

$$ M_1 = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} $$

$$ M_2 = \begin{bmatrix} b_{11} & b_{12} & \cdots & b_{1p} \\ b_{21} & b_{22} & \cdots & b_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ b_{n1} & b_{n2} & \cdots & b_{np} \end{bmatrix} $$

$$ M_1 \cdot M_2 = \begin{bmatrix} a_{11}b_{11} +\cdots + a_{1n}b_{n1} & a_{11}b_{12} +\cdots + a_{1n}b_{n2} & \cdots & a_{11}b_{1p} +\cdots + a_{1n}b_{np} \\ a_{21}b_{11} +\cdots + a_{2n}b_{n1} & a_{21}b_{12} +\cdots + a_{2n}b_{n2} & \cdots & a_{21}b_{1p} +\cdots + a_{2n}b_{np} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1}b_{11} +\cdots + a_{mn}b_{n1} & a_{m1}b_{12} +\cdots + a_{mn}b_{n2} & \cdots & a_{m1}b_{1p} +\cdots + a_{mn}b_{np} \end{bmatrix} $$

Now if you are looking to calculate the position of an element in the matrix \(M_{3}\), follow the steps below:

  • Look in which row and column the element lies
  • After knowing this, select that row from the first matrix \(M_{1}\) and that column from the second matrix \(M_{2}\)
  • After you choose row and column, multiply each and every entity present in them one by one
  • Among these entities, your desired element value also lie that can be determined instantly

Besides that, the source of calculator-online designed a free online matrix calculator to determine any element’s position in the matrix.

Main Conditions of Matrix Multiplication:

So, how to do matrix multiplication if the numbers are complex? It’s quite simple as we are going to discuss the following steps that will help you to resolve such problems as well. These include:

  • The number of columns in the first matrix must be equal to the number of rows in the second matrix
  • After multiplication, the final matrix will contain rows equal to first matrix and columns equal to the second matrix
  • For example; if you find the product of a matrix of order ‘n’ by ‘k’ with other matrix of order ‘k’ by ‘m’, the order of the final matrix will be ‘n’ by ‘m’

This may confuse you a little bit but we are going to clear it with the help of following matrices below:

$$ \begin{bmatrix}10 & 10 \\ 8 & 8 \\\end{bmatrix} \hspace{0.25in} \begin{bmatrix}9 \\ 5 \\\end{bmatrix} $$

Now if you see both of these matrices, you will clearly see that the first matrix has two columns and the second matrix has two rows. As they fulfill the condition, they are perfect for multiplication. Now when you will multiply them, you will get the following matrix:

$$ \begin{bmatrix}140 \\ 112 \\\end{bmatrix} $$

Now if you check its order, it is 2 by 1 which indicates that its rows are equal to the first matrix and columns are equal to the second matrix.

Moreover, you can speed up your calculations by using our best matrix multiplication calculator.

Properties of Matrix Multiplication:

Multiplication of the matrices posses frequent properties that are enlisted as follows:

Commutative Property:

Matrix multiplication does not hold the commutative property.


Associative Property:

Matrices multiplication follows the associative law of product:


Distributive Property:

A(B+C) = AB +AC                     Left Distributive Law
(A+B)+C = AC+BC                   Right Distributive Law

These distributive laws are also satisfied by real numbers that could also be verified by using distributive property calculator

Identity Property:

If we multiply any matrix with the identity matrix, we will get the same matrix always.

IA = A or AI = A

Multiplicative Property With Zero:

If we multiply the matrix with the zero matrix(a matrix whose all entities are zero), we will get the zero matrix.

AO = OA= O

How To Multiply Matrices?

Let us resolve an example so that you may understand the matrices multiplication properly. Stay focused!

Example # 01:

How to multiply a matrix with the identity matrix given below:

$$ \begin{bmatrix} 5 \\ 4 \\\end{bmatrix} $$


As the given matrix has one column only, so the identity matrix must also contain only one row and is as follows:

$$ \begin{bmatrix}1 & 0 \\\end{bmatrix} $$

Performing Matrices Multiplication:

$$ \begin{bmatrix} 5 \\ 4 \\\end{bmatrix} \cdot \begin{bmatrix}1 & 0 \\\end{bmatrix} $$

$$ \begin{bmatrix} ( 5*1 ) ( 5*0 ) \\ ( 4*1 ) ( 4*0 ) \\\end{bmatrix} $$

$$ \begin{bmatrix}(5 ) (0 ) \\ (4 ) (0 ) \\\end{bmatrix} $$

$$ \begin{bmatrix} 5 & 0 \\ 4 & 0 \\\end{bmatrix} $$

No doubt that manual matrix calculations look daunting, the use of the free multiply matrices calculator makes great sense here.

This may be time consuming for you. That is why you should also make use of the free multiply matrices calculator.

How Matrix Multiplication Calculator Works?

Allow this free matrix multiplier to determine the product of two matrices that are perfect for multiplication. Let us move on to learn its usage!


  • First of all, select the number of rows and columns for the first matrix
  • Now do the same for the second matrix. But keep in mind that its number of rows must be equal to the number of columns of the first matrix
  • Now tap the “set matrices” to get the desired matrices layouts
  • After you get the layouts, enter all the values for both of the matrices
  • Tap the calculate button


The free multiplying matrices calculator does the following calculations:

  • Determines matrices multiplication
  • Shows step by step calculations of steps involved


How to multiply matrices 2×2 instantly?

If you are looking for the immediate product of these matrices, make use of our free online matrix multiplication calculator.

Is it possible to multiply the matrices that have the following order: 2 by 3 and 4 by 3

No, the multiplication is not possible. This is because the number of columns of the first matrix is not equal to the number of rows of the second matrix.

What is the order of the matrix multiplication?

Suppose you are about to multiply two matrices that satisfy the product conditions. You will always start from the most left entity and forward to the right one. So the order of matrix multiplication is always from left to right that could also be obtained by using a free online matrix multiplication calculator.

What is matrix scalar multiplication?

In scalar multiplication, you just take one number that is a scalar and multiply it with each and every entity of the matrix with which it is supposed to get the product.

What other calculators can I use for various matrix calculations?

We have designed various matrix calculators as this is the basis of the algebra. You can subject to the calculators below to determine various factors with our matrix related calculators:


So we understood all the basics of matrix products in the read, we hope you may not feel difficulty in using the matrix multiplication calculator to determine the results.


From the source of Wikipedia: Matrix multiplication, Fundamental applications, General properties, Square matrices

From the source of khan academy: Zero and identity matrices, Strategies, Real-life Applications

From the source of lumen learning: Introduction to Matrices, Scalar Multiplication, Matrix Multiplication