Matrix Multiplication Calculator: Multiply Now

This is a free online tool that can be used to quickly and accurately multiply matrices together. Simply input the values within your matrix and the calculator will work out the product for you. This can be useful for students, teachers, engineers, researchers and professionals.

Matrix multiplication is a key concept which is used across many areas including algebra, linear algebra, computer science, physics, engineering, statistics and data science. It can be commonly used to solve systems of linear equations, geometric transformations, digital image manipulation, analysing data sets and creating machine learning models.

A matrix is simply an array of numbers organised in rows and columns. When multiplying two matrices together, the number of columns in the first matrix must equal the number of rows in the second matrix. For example, if the first matrix was 2 × 3 and the second matrix was 3 × 2, you would end up with a 2 × 2 matrix.

Using our online Matrix Multiplication Calculator will help you to quickly multiply matrices. Use this calculator to multiply 2 × 2 matrices, 3 × 3 matrices and much more.

How to Use the Matrix Multiplication Calculator

To use this calculator simply follow the steps below:
Choose how many rows and columns you want the first and second matrices to have.

Enter the values for the first matrix.

Enter the values for the second matrix carefully.

Ensure that the number of columns in the first matrix equal the number of rows in the second matrix.

Click calculate.

The calculator will display your resulting matrix.

Clear the values to calculate another matrix multiplication problem.

If you have 2 rows and 3 columns in your first matrix, you will need 3 rows in your second matrix. The resulting matrix will then have 2 rows and however many columns the second matrix had.

Matrix Multiplication Formula

When multiplying matrices together, there is a specific formula to follow. To calculate each value in the resulting matrix, you multiply the corresponding elements in a row from the first matrix by the corresponding elements in a column from the second matrix. Then add them together.

Understanding your Result

Once you have inputted your matrix values, the calculator will show you a new matrix which represents the product of the two matrices.

To understand what your result means, you will need to check:

The dimensions of the matrix. The number of rows will come from the first matrix and the number of columns will come from the second matrix.

Each element in the resulting matrix represents the dot product of a row from the first matrix and a column from the second matrix.

If you get positive values, this means that the sum of the calculations for that row and column were positive.

If you get negative values, it could be due to having negative values in the original matrices.

If you get a zero value, this could be because the positive and negative values have cancelled each other out.

It is important to remember that matrix multiplication is generally not commutative meaning you might get a different result depending on the

order you multiply the matrices in.

In some circumstances you may not be able to reverse the order due to the dimensions not matching.

What Affects the Result?

There are a few things which will affect the outcome of your matrix multiplication.

Matrix Dimensions

The dimensions of your matrices will affect whether you can multiply them. The number of columns in matrix A must equal the number of rows in matrix B.

Values Entered

All values entered into the matrices will affect the outcome of the resulting matrix. If you change one value in the matrices, it could affect multiple values in the resulting matrix.

Positive and Negative Values

You can have positive values, negative values and zero values in your matrices. This will affect the signs and values of your resulting matrix.

Order of Multiplication

As matrix multiplication is generally not commutative, you may get a different result if you multiply A by B or B by A.

Decimal Values

If your matrices have decimal values, you may get a result with several decimal places. This will depend on how the calculator displays numbers.

Incorrect Values Entered

If you enter an incorrect value, or put the value in the wrong row/column or chose the wrong dimensions, you may get an incorrect result.

Advantages and Uses of Matrix Multiplication

There are many advantages to using a Matrix Multiplication Calculator.

Saves Time

This calculator will save you time as it will automatically carry out multiple multiplications and additions.

Minimises Error

This calculator will help reduce error in your calculations.

Helps with Understanding

Students can compare their manual calculations to the calculator to understand the rules of matrix multiplication.

Useful for Linear Algebra

This calculator can help you solve problems relating to matrices such as transformations, equations and models.

Engineering

Engineers use matrix multiplication for structural analysis, electrical systems, robotics and mechanical modelling.

Computer Graphics

Matrix multiplication is often used in computer graphics to rotate, scale, translate and transform objects.

Data Science

Matrices can be used to organise data sets and carry out calculations in machine learning algorithms, neural networks and statistical models.

Physics

Researchers will use matrix multiplication to show mathematical relationships.

Accuracy, Assumptions and Limitations

As long as you enter the correct values and dimensions into the calculator, you will get an accurate result. The calculator will follow the standard rules of matrix multiplication and calculate each value in the resulting matrix by multiplying the rows by the columns.

There are a few assumptions and limitations to bear in mind:
The dimensions of the matrices must be compatible. The number of columns in matrix A must equal the number of rows in matrix B.

The accuracy of the result will depend on the values entered.

Results with decimal values may be rounded.

Very large matrices may take longer to calculate.

The order of the matrices can affect the result. Switching the order may give you a different result or you may not even be able to switch the order.

Some matrices have special properties such as identity matrices, zero matrices, diagonal matrices and square matrices.

If you need very accurate results for scientific or engineering calculations, you should consider the accuracy of the calculator and use an appropriate maths package to confirm any important results.

This calculator has been designed to make standard matrix multiplications easier. It will not automatically tell you about all the advanced properties of matrices and will not solve any other matrix operations.

FAQs

What is a Matrix Multiplication Calculator?

A Matrix Multiplication Calculator is an online tool which can be used to multiply two matrices together. It will carry out the necessary calculations to multiply rows by columns and add them together. This saves time and reduces error in your calculations.

Can I multiply matrices with different dimensions?

Yes, you can multiply matrices with different dimensions as long as the number of columns in the first matrix equals the number of rows in the second matrix. For example, a 2 × 3 matrix can be multiplied by a 3 × 4 matrix to get a 2 × 4 matrix.

Is matrix multiplication commutative?

No, matrix multiplication is generally not commutative. This means that if you multiply matrix A by matrix B, you may not get the same result as if you multiplied matrix B by matrix A. You might not even be able to switch the order.

How do I calculate a matrix multiplication for 3 × 3 matrices?

To multiply two 3 × 3 matrices together, you need to multiply each row in the first matrix by each column in the second matrix. Then add together the three products to calculate each element in the resulting matrix. Alternatively, you can input both matrices into the online calculator.

Where is matrix multiplication used in real life?

Matrix multiplication can be used in computer graphics, engineering, robotics, physics, economics, data science, machine learning and image processing. It can help carry out mathematical transformations, organise information and solve complex problems.