Enter two matrices and instantly get calculation results with detailed step-by-step explanations. Every step is explained in clear, easy-to-understand language to help you master matrix multiplication principles.
Professional online matrix multiplication calculator supporting any matrix size calculations with detailed step-by-step explanations and visual demonstrations. Perfect for students learning linear algebra and engineers working with matrix operations.
Online Matrix Multiplication Calculator | Step-by-Step Solutions | Visual Demonstrations | Any Matrix Size | Free to Use
Follow these simple steps to calculate matrix multiplication and understand the detailed process with our interactive calculator
Use the dropdown menus to select the number of rows and columns for Matrix A and Matrix B. The calculator automatically ensures compatibility.
Click on each cell and enter numbers. You can use integers, decimals, or negative numbers. Press Tab to move to the next cell quickly.
The calculator shows a green checkmark when matrix dimensions are compatible (A columns = B rows). Red warning appears for incompatible sizes.
Click "Start Calculation" to get instant results with detailed step-by-step explanations showing how each element is calculated.
Performs matrix multiplication and shows detailed step-by-step solutions. Only enabled when matrices are compatible and contain valid data.
Automatically fills both matrices with example data so you can immediately see how the calculator works and explore the results.
Resets both matrices to zero values and clears all results. Use this to start fresh with new matrix calculations.
Understand the mathematical definition, requirements, calculation rules, and practical significance of matrix multiplication with comprehensive explanations
Matrix multiplication is a fundamental operation in linear algebra used to calculate the product of two matrices. Given matrices A and B, their product C = A × B is defined as follows:
Mathematical Expression:
C[i,j] = Σ(k=1 to n) A[i,k] × B[k,j]
where i represents row, j represents column, k is the summation variable
Simply put: The element at row i, column j of result matrix C equals the sum of products between row i of matrix A and column j of matrix B.
Core Rule: The number of columns in the first matrix must equal the number of rows in the second matrix
Check if the number of columns in matrix A equals the number of rows in matrix B. If not equal, matrix multiplication cannot be performed.
The result matrix C has dimensions: A's rows × B's columns. Example: (3×2) × (2×4) = (3×4)
For each position (i,j) in the result matrix, calculate the dot product of row i from A and column j from B.
Dot product = a₁×b₁ + a₂×b₂ + ... + aₙ×bₙ, where n is the number of columns in A (also the number of rows in B).
O(m×n×p)
m,n,p are matrix dimensions
O(m×p)
Storage space for result matrix
m×n×p multiplications
m×p×(n-1) additions
Generally: A×B ≠ B×A
(A×B)×C = A×(B×C)
A×(B+C) = A×B + A×C
k×(A×B) = (k×A)×B = A×(k×B)
Through detailed step-by-step explanations, gain complete understanding of matrix multiplication calculation principles
First, confirm that the number of columns in the first matrix equals the number of rows in the second matrix. Only when this condition is met can the two matrices be multiplied.
For each position in the result matrix, take the corresponding row from the first matrix and the corresponding column from the second matrix to perform calculations.
Repeat step 2 until all elements of the result matrix are calculated. Our tool displays the detailed process of each step.
Click example cards to automatically fill the calculator and instantly experience matrix multiplication calculations of different sizes
Most basic matrix multiplication example, perfect for beginners to understand fundamental concepts
Each element is the result of corresponding row-column multiplication and summation
Medium complexity matrix multiplication with more calculation steps
Requires calculating 9 elements, each containing 3 multiplications
Different sized matrices multiplication, demonstrating size compatibility rules
Result matrix size is 2×2
Example data will be automatically filled into the calculator above, and you can immediately click "Start Calculation" to view detailed steps
Learn how to perform matrix multiplication with our comprehensive guide, from basic concepts to advanced techniques
Remember: Columns of first matrix = Rows of second matrix
Verify that matrix A's columns equal matrix B's rows
Create result matrix C with dimensions (A rows × B columns)
For each position (i,j), compute row i × column j
Multiply corresponding elements and sum the results
Understand the matrix multiplication algorithm, implementation methods, and optimization techniques used in programming
import numpy as np # Create matrices A = np.array([[1, 2], [3, 4]]) B = np.array([[5, 6], [7, 8]]) # Matrix multiplication C = np.dot(A, B) # or A @ B print(C) # [[19 22], [43 50]]
def matrix_multiply(A, B):
rows_A, cols_A = len(A), len(A[0])
rows_B, cols_B = len(B), len(B[0])
if cols_A != rows_B:
raise ValueError("Invalid dimensions")
C = [[0] * cols_B for _ in range(rows_A)]
for i in range(rows_A):
for j in range(cols_B):
for k in range(cols_A):
C[i][j] += A[i][k] * B[k][j]
return CAdvanced algorithm that reduces time complexity from O(n³) to O(n^2.807) for large matrices.
Matrix multiplication can be parallelized for better performance on multi-core systems.
Discover the important applications of matrix multiplication across various fields and industries
In 3D graphics rendering, matrix multiplication is used for coordinate transformations, rotations, scaling, and other operations. Every 3D object transformation requires matrix multiplication calculations.
Matrix multiplication is the core computation in neural networks. From simple linear regression to complex deep learning models, matrix operations are extensively used.
In statistical analysis and data science, matrix multiplication is used for data transformations, correlation analysis, principal component analysis, and other important calculations.
Physics simulations, engineering calculations, economic modeling, and other fields all require matrix multiplication to solve complex mathematical problems.
Master the mathematical formula behind matrix multiplication and understand its theoretical foundation
Where:
Common questions and answers about the matrix multiplication calculator
A:Matrix multiplication has strict size requirements: the number of columns in the first matrix must equal the number of rows in the second matrix. For example, a 3×2 matrix can only be multiplied with a 2×n matrix, not with a 3×n matrix.
A:No, it's not. Generally, A×B ≠ B×A. Even when both matrices are square and can be multiplied in either order, the results are usually different. This is an important distinction from regular number multiplication.
A:You can check if the result matrix dimensions are correct (should be first matrix rows × second matrix columns), then verify the calculation process of a few key elements. Our tool provides detailed steps to help with verification.
A:Our calculator supports matrices up to 5×5, which meets most learning and basic application needs. For larger matrices, we recommend using professional mathematical software.
A:Detailed step explanations help you understand the principles of matrix multiplication, not just get the answer. This is very helpful for learning linear algebra, exam preparation, or understanding related algorithms.
A:You can use your browser's screenshot function to save results, or copy the text content of step explanations. We're considering adding PDF export and sharing link features.
Experience our interactive matrix multiplication calculator immediately and get detailed step-by-step explanations with visual demonstrations. Completely free, no registration required, supports any matrix size calculations.
Already helped thousands of students and engineers understand matrix multiplication principles and obtain accurate calculation results.