Matrix Multiplication Calculator (2024)

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:

  • To determine the determinant of any matrix, tap the determinant calculator
  • To find the eigenvalue of any matrix, tap the eigenvalue calculator.
  • If you are interested in determining the null space matrix, try using null space calculator


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

Matrix Multiplication Calculator (2024)


How do you know if a matrix multiplication is possible? ›

You can only multiply matrices if the number of columns of the first matrix is the same as the number of rows as the second matrix. For example, say you want to multiply A x B. If A is a 3x1 matrix, B has to be a 1xY matrix (Y can be any number), because A only has 1 column.

Does photomath do matrices? ›

Calculus. Matrices, derivatives, integrals, oh my! From pre-calculus to calculus, we can still help you when you reach these advanced topics. This is often when it's really helpful to see theorems and rules explained in context — so don't worry; we do that, too.

What is the most efficient way to multiply matrices? ›

In linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm for matrix multiplication. It is faster than the standard matrix multiplication algorithm for large matrices, with a better asymptotic complexity, although the naive algorithm is often better for smaller matrices.

What is the website to solve matrix problems? › is the most convenient free online Matrix Calculator. All the basic matrix operations as well as methods for solving systems of simultaneous linear equations are implemented on this site.

What makes matrix multiplication not possible? ›

Because we can only multiply matrices if their dimensions are compatible. Matrices are only compatible if the number of columns in the first matrix equals the number of rows in the second column.

Which matrix multiplication would not be possible? ›

The matrix multiplication is not possible in c2*2 with 3*3 and d5*6 with 3*3. The matrix multiplication is possible when the number of columns in the first matrix is equal to the number of rows in the second matrix and vice versa.

Does Photomath count as cheating? ›

stop saying it's for cheating

This app is excellent because it shows you how to do the equation so you don't have to use the app when you see a question like that. If your child uses it to cheat, then let them. They'll suffer the consequences of not knowing the content in an actual exam and step up later.

What app solves matrices? ›

Linear Algebra - Matrix Solver 4+

This app is designed for students and engineers who use operations with matrices and vectors in their studies or work. The program, using the necessary formulas, will perform step-by-step calculations and display a detailed solution.

Does Photomath give accurate answers? ›

Photomath is using cutting edge AI-powered solving and recognition capabilities, that is why most of our solutions are to the point, and the issue might be in the incorrectly scanned problem. If the result is wrong, first check that the math problem you scanned is the same as the one recognized by Photomath.

Why is matrix multiplication slow? ›

Matrix operations may not be superior if the matrices are sparse. In that case a loop that avoids unnecessary operations can be much faster than a dense matrix operation that wastes a lot of time on multiplying and adding zeros. It looks like you're using the element wise product .

What is important for matrix multiplication? ›

For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB.

How to improve matrix multiplication? ›

The remedy is a technique known as copy optimization. Before starting matrix multiplication, the matrices are copied into memory locations, the first matrix being stored in row-major format block-wise, while the second matrix is stored in column major format block-wise. This also helps in achieving spatial locality.

Can you solve a matrix on a calculator? ›

Use the MATRIX Mode to perform calculations involving matrices of up to 3 rows by 3 columns. To perform a matrix calculation, you first assign data to special matrix variables (MatA, MatB, MatC), and then use the variables in the calculation as shown in the example below. 1. Press (MATRIX) to enter the MATRIX Mode.

What is the code for matrix? ›

What is matrix code? Matrix code refers to a two-dimensional barcode that stores information in a pattern of black and white squares.

Can you multiply a 2x3 and 2x3 matrix? ›

Explanation: For matrix multiplication AB , the number of columns of A must be equal to the number of rows of B. A has 3 columns and B has 2 rows, so they are not compatible for multiplication.

Can a 2x2 and 2x3 matrix be multiplied? ›

No, we cannot multiply a 2x3 and 2x2 matrix because for multiplying matrices, two matrices should be compatible. Since the number of columns in the first matrix(3) is not equal to the number of rows in the second matrix(2), we cannot perform matrix multiplication for this case.

How do you tell if a matrix can be solved? ›

In order to identify the solution/s of a matrix system AX=B A X = B , you should find the rank of the augmented matrix A:B and that of the coefficient matrix A . If both ranks are equal, then the system possesses at least one solution.

How to tell if matrix multiplication is undefined? ›

The product of two matrices is undefined whenever the rows of the first matrix (reading right to left) do not match the column of the second matrix.

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