01. What is the main rule for multiplying two matrices?

The necessary condition for existence is that the number of columns of Matrix A must be exactly equal to the number of rows of Matrix B. If this rule is not met, the multiplication is mathematically impossible.

02. Is matrix multiplication commutative (A × B = B × A)?

No. Unlike multiplication of common numbers, in linear algebra the order of factors changes the product. Multiplying Matrix A by Matrix B usually generates a completely different result than multiplying Matrix B by Matrix A.

03. How is the step-by-step calculation of matrix multiplication done?

The calculation is not done element by element. You must multiply the elements of an entire row of the first matrix by the corresponding elements of a column of the second matrix, and then sum all these products to find a single value in the resulting matrix.

Matrix Multiplier

Matrix A

Rows

Columns

Determinant: 0 (Singular)

Matrix B

Rows

Columns

Determinant: 0 (Singular)

Result of Matrix Multiplication

How to use the Matrix Multiplier

To use the Matrix Multiplier, follow the steps below:

Enter the size of the first matrix (Matrix A) in the Rows and Columns fields.
Enter the desired values in Matrix A in the corresponding fields, which can be integers or decimals. (Fractions are also allowed, such as 3/4)
Enter the size of the second matrix (Matrix B) in the Rows and Columns fields.
Enter the desired values in Matrix B in the corresponding fields, which can be integers or decimals. (Fractions are also allowed, such as 3/4)

Remember that the number of columns of Matrix A must be equal to the number of rows of Matrix B for multiplication to be possible.

The multiplier will provide the result of multiplying the matrices, allowing you to clearly visualize the product of the two matrices.

Need to add or subtract matrices with the same dimensions? Use the basic matrix operations calculator.

Example of Matrix Multiplication

Suppose you have two matrices, A and B, and you want to calculate their product.

When you enter the values of matrices A and B in the corresponding fields, you will see the result of multiplying the matrices, allowing you to clearly visualize the product of the two matrices. As shown in the image below:

This example demonstrates how the matrix multiplier can be a useful tool for performing matrix multiplication calculations quickly and accurately, especially in contexts such as linear algebra, computer graphics, and data analysis.

Defining Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that combines two matrices to produce a third matrix. Unlike basic operations, matrix multiplication involves summing the products of corresponding elements from rows of the first matrix and columns of the second matrix.

Since matrix multiplication combines elements from two matrices, it has an existence condition: it is only possible when the number of columns of the first matrix is equal to the number of rows of the second matrix. If this condition is not met, matrix multiplication cannot be performed.

Properties of Matrix Multiplication

Associative: (AB)C = A(BC) for any matrices A, B and C, provided the dimensions are compatible for multiplication.
Distributive: A(B + C) = AB + AC and (A + B)C = AC + BC for any matrices A, B and C, provided the dimensions are compatible for multiplication.
NNon-commutative: AB ≠ BA for any matrices A and B. This means that the order of matrices in multiplication is important and can affect the result.
Multiplicative Identity Property: There exists an identity element in matrix multiplication, which is the identity matrix. Multiplying any matrix by the identity matrix does not change the original matrix. That is, for any matrix A, we have AI = IA = A, where I is the identity matrix of compatible dimensions with A.
Multiplicative Null Element Property: There exists a null element in matrix multiplication, which is the null matrix. Multiplying any matrix by the null matrix results in the null matrix. That is, for any matrix A, we have A0 = 0A = 0, where 0 is the null matrix of compatible dimensions with A.
Dimension Property: Matrix multiplication respects the dimensions of the involved matrices. That is, if A is an m × n matrix and B is an n × p matrix, then their product AB will be an m × p matrix.

This operation is essential in various fields, including physics, engineering, computer science, and data analysis. It is used to solve systems of linear equations, perform transformations in computer graphics, and analyze data in machine learning, among many other applications.

Formulas for Matrix Multiplication

The formula for multiplying two matrices A and B, where A is an m × n matrix and B is an n × p matrix, is given by:

\mathbf{C_{i,j} = \sum_{k=1}^{n} A_{i,k} \times B_{k,j}}

C_i,j Element of the resulting matrix C at position (i, j).

A_i,k Element of matrix A at position (i, k).

B_k,j Element of matrix B at position (k, j).

n Number of columns of A (equal to the number of rows of B).

Although it may seem complex, matrix multiplication is an operation that can be performed systematicall by following the formula above. With practice, you will be able to perform matrix multiplications quickly and accurately, which is especially useful in contexts such as linear algebra, computer graphics, and data science.

DECMATRIX

Matrix Multiplier

Matrix A

Matrix B