Matrix Transpose & Multiplication Calculator – 3×3 and 4×4 in 2 Easy Steps
Matrix Transpose & Multiplication
Free matrix transpose and multiplication calculator for 3×3 and 4×4 matrices.
Matrix A
Matrix B
Result
Use this free matrix transpose and multiplication calculator to multiply two matrices or flip a matrix in one click.
Quick answer
To multiply matrices, each entry of A × B is the dot product of a row of A with a column of B. To find the matrix transpose, swap rows and columns: the entry in row i, column j moves to row j, column i. Example: A = [[1,2,3],[4,5,6],[7,8,9]] and B = [[9,8,7],[6,5,4],[3,2,1]] give A × B = [[30,24,18],[84,69,54],[138,114,90]], and Aᵀ = [[1,4,7],[2,5,8],[3,6,9]].
Table of Contents
Matrix multiplication and transposition are the bread and butter of linear algebra – and the source of endless arithmetic slips when done by hand. A single wrong product in a 4×4 multiplication means 16 entries to recheck.
Our free calculator handles 3×3 and 4×4 square matrices. Choose Multiply (A × B), Transpose A or Transpose B, enter the values and get the result grid instantly.
How Matrix Multiplication Works
For C = A × B, each entry cᵢⱼ = (row i of A) · (column j of B). For the example above, c₁₁ = 1·9 + 2·6 + 3·3 = 9 + 12 + 9 = 30, and c₂₂ = 4·8 + 5·5 + 6·2 = 32 + 25 + 12 = 69.
- Matrix multiplication is not commutative: A × B is usually different from B × A.
- It is associative: (A × B) × C = A × (B × C).
- For square matrices of size n, there are n² entries, each needing n multiplications.
How the Matrix Transpose Works
The matrix transpose Aᵀ flips A over its main diagonal: rows become columns. The diagonal stays the same. Useful rules:
| Property | Rule |
|---|---|
| Double transpose | (Aᵀ)ᵀ = A |
| Sum | (A + B)ᵀ = Aᵀ + Bᵀ |
| Product | (A × B)ᵀ = Bᵀ × Aᵀ |
| Symmetric matrix | A = Aᵀ |
How to Use the Matrix Transpose & Multiplication Calculator
Get your result in two steps.
Step 1: Choose Size and Operation, Then Enter Values
Select the Matrix Dimension (3 × 3 or 4 × 4) and the Operation: Multiply (A × B), Transpose A or Transpose B. Fill in Matrix A and Matrix B.
Step 2: Calculate and Read the Result
Click Calculate. The Result grid shows A × B or the matrix transpose you chose. Click Reset to clear all cells and start a new problem.
Where Matrix Multiplication and Transpose Are Used
- 3D graphics and games: rotation, scaling and camera transforms use 4×4 matrices.
- Machine learning: neural networks multiply weight matrices by input vectors millions of times.
- Statistics: regression formulas such as (XᵀX)⁻¹Xᵀy use transpose, multiplication and inverse.
- Engineering: systems of equations and structural analysis.
Common Mistakes
- Multiplying entry by entry instead of row by column.
- Assuming A × B = B × A.
- Forgetting that (AB)ᵀ reverses the order: BᵀAᵀ.
Need the inverse or eigenvalues next? Use the Matrix Inverse Calculator or the Eigenvalue Eigenvector Calculator.
Frequently Asked Questions
How do you multiply two matrices?
Each entry of the result is the dot product of a row from the first matrix and a column from the second.
What is a matrix transpose?
It is the matrix you get by swapping rows and columns, so entry (i, j) becomes (j, i).
Is A × B the same as B × A?
Usually not. Matrix multiplication is not commutative.
What is the transpose of a product?
(A × B)ᵀ = Bᵀ × Aᵀ – the order is reversed.
Which sizes does the calculator support?
Square 3 × 3 and 4 × 4 matrices.
Is the calculator free?
Yes. It is free, runs in your browser and needs no sign-up.