Matrix Operations
Matrices are fundamental in linear algebra, computer graphics, machine learning, and physics.
Key Operations
| Operation | Rule |
|---|---|
| Addition | Element-wise, same dimensions |
| Multiplication | Row × Column dot products |
| Determinant (2×2) | ad - bc |
| Transpose | Swap rows and columns |
How to Use This Matrix Calculator
Enter matrix dimensions and values. Select an operation: addition, subtraction, multiplication, transpose, determinant, or inverse. The calculator shows step-by-step results.
Formula & How It Works
Addition: element-by-element (same dimensions required). Multiplication: C[i,j] = Σ A[i,k] × B[k,j]. Determinant (2×2): ad – bc. Inverse: A⁻¹ = adj(A) / det(A).
Calculation Example
[1 2; 3 4] × [5 6; 7 8] = [19 22; 43 50]. Determinant of [1 2; 3 4] = 1×4 – 2×3 = -2.
Expert Tips
Matrix multiplication is not commutative: A×B ≠ B×A. The inverse exists only when the determinant ≠ 0. Matrices are fundamental in computer graphics (transformations), machine learning (neural networks), and physics.
Frequently Asked Questions
When is matrix multiplication defined?
A(m×n) × B(n×p) works when A columns = B rows. Result is m×p.
What does the determinant mean?
The determinant represents scaling factor of the linear transformation. det=0 means the matrix is singular (not invertible).
What is the transpose?
Transpose flips rows and columns. Element a_ij becomes a_ji.