Matrix Calculator
Perform matrix operations: addition, subtraction, scalar multiplication, and matrix multiplication for 2×2 and 3×3 matrices. Find transpose and trace of a matrix.
∑ Linear Algebra📐 C = A × B (matrix multiplication)🔢 Math
Operation
A row 1: a₁₁
a₁₂
A row 2: a₂₁
a₂₂
B row 1 / scalar k: b₁₁
b₁₂
B row 2: b₂₁
b₂₂
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Matrix Calculator | — | C = A × B (matrix multiplication) | matrix |
Step-by-Step Examples
Example 1
Matrix Multiplication
A=[[1,2],[3,4]] × B=[[5,6],[7,8]].
- C₁₁=1×5+2×7=19
- C₁₂=1×6+2×8=22
- C₂₁=3×5+4×7=43
- C₂₂=3×6+4×8=50
- Result: [[19,22],[43,50]]
✓ [[19,22],[43,50]]
Example 2
Addition
[[1,2],[3,4]] + [[5,6],[7,8]].
- Add element-wise
- [[6,8],[10,12]]
✓ [[6,8],[10,12]]
Example 3
Transpose
Transpose [[1,2],[3,4]].
- Rows become columns
- [[1,3],[2,4]]
- Trace stays same: 1+4=5
✓ [[1,3],[2,4]]
Real-World Applications
Data Science
Matrix operations are the foundation of machine learning (weights, transformations).
Computer Graphics
3D rotations, scaling, and projections are matrix multiplications.
Engineering
Systems of equations solved with matrix methods (Gaussian elimination).
Statistics
Covariance matrices, PCA, and regression use matrix algebra.
Common Mistakes to Avoid
⚠️
Matrix multiplication is not commutative
AB ≠ BA in general. Always multiply in the correct order.
⚠️
Dimension mismatch
A×B requires A's columns = B's rows. (m×n)(n×p) = (m×p). Mismatched dimensions: undefined.
⚠️
Addition requires same dimensions
You can only add matrices of exactly the same size.
Frequently Asked Questions
What is a matrix? ▾
A rectangular array of numbers arranged in rows and columns. Used to represent linear transformations, systems of equations, and data.
Why is matrix multiplication done that way? ▾
Each element cᵢⱼ = dot product of row i of A with column j of B. This corresponds to composing linear transformations.
What is the identity matrix? ▾
The matrix I where AI = IA = A. For 2×2: [[1,0],[0,1]]. It is the matrix equivalent of the number 1.
What is the trace? ▾
Sum of diagonal elements. Trace(A) = a₁₁ + a₂₂ + ... Equals the sum of eigenvalues.
What is a rotation matrix? ▾
[[cos θ, −sin θ],[sin θ, cos θ]] rotates vectors by angle θ counterclockwise. Widely used in graphics and robotics.
Related Math Calculators
Formula Explorer connections
Interpretation: This relationship represents vectors, linear systems or transformations using matrix structure and vector operations. Assumption: Matrix dimensions must be compatible. Inversion, decomposition and unique solutions require rank, conditioning or nonzero-determinant conditions.