Matrix Operations
Matrices store linear transformations. Add/subtract element-wise. Scalar multiply: scale each entry. Transpose: swap rows and columns. Inverse A⁻¹ satisfies AA⁻¹=I. 2×2 inverse: swap diagonal, negate off-diagonal, divide by det.
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Matrix addition requires same dimensions. A is invertible iff det(A)≠0 (nonsingular). Symmetric: A=Aᵀ. Diagonal: nonzero only on diagonal.
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Why: Matrices represent linear maps, solve systems, and model transformations. Inverse solves Ax=b via x=A⁻¹b. Transpose appears in least squares and covariance.
How: Addition: (A+B)ᵢⱼ=aᵢⱼ+bᵢⱼ. Transpose: (Aᵀ)ᵢⱼ=aⱼᵢ. 2×2 inverse: [[a,b],[c,d]]⁻¹=(1/(ad−bc))[[d,−b],[−c,a]]. Only square matrices with det≠0 have inverses.
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Matrix A
Matrix B
Entry Magnitudes
Positive vs Negative vs Zero
Calculation Steps
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
2×2: [[a,b],[c,d]]⁻¹ = (1/(ad−bc))[[d,−b],[−c,a]].
— Formula
det(A)=0 → A singular, no inverse.
— Singular
Matrix Operations Rules
- • Addition/Subtraction: Defined only when both matrices have the same dimensions (m×n). Add or subtract corresponding elements.
- • Scalar multiplication: Multiply every entry by the scalar k. (kA)ᵢⱼ = k × aᵢⱼ
- • Transpose: Swap rows and columns. (Aᵀ)ᵢⱼ = aⱼᵢ. A matrix is symmetric if A = Aᵀ.
- • Inverse: Only square matrices can have inverses. A⁻¹ exists iff det(A) ≠ 0. A·A⁻¹ = I (identity).
- • Identity matrix I: Square matrix with 1s on the diagonal, 0s elsewhere. A·I = I·A = A.
Applications
Computer Graphics
3×3 and 4×4 matrices represent rotations, scaling, and translations in 2D/3D rendering.
Systems of Equations
Ax = b. Matrix inverse gives x = A⁻¹b when A is square and invertible.
Data Science
Covariance matrices, PCA, and linear transformations use matrix algebra.
Physics & Engineering
Quantum mechanics, circuit analysis, and structural mechanics rely on matrices.
Matrix Properties
| Property | Definition |
|---|---|
| Symmetric | A = Aᵀ (rows = columns when transposed) |
| Diagonal | Non-zero only on main diagonal |
| Identity | I with 1s on diagonal, 0s elsewhere |
| Singular | det(A) = 0, no inverse exists |
Frequently Asked Questions
When can I add or subtract two matrices?
Only when they have the same dimensions (same number of rows and columns). Element-wise addition: (A+B)ᵢⱼ = aᵢⱼ + bᵢⱼ.
What is the identity matrix?
The n×n identity matrix I has 1s on the main diagonal and 0s elsewhere. For any matrix A, A·I = I·A = A.
When does a matrix have an inverse?
A square matrix A has an inverse A⁻¹ iff det(A) ≠ 0. Then A·A⁻¹ = A⁻¹·A = I.
What does transpose mean?
Transpose Aᵀ swaps rows and columns: (Aᵀ)ᵢⱼ = aⱼᵢ. A symmetric matrix satisfies A = Aᵀ.
How is matrix algebra used in computer graphics?
3×3 and 4×4 matrices represent rotations, scaling, shearing, and translations. Composing transformations = multiplying matrices.
What is scalar multiplication?
Multiply every entry by a constant k: (kA)ᵢⱼ = k·aᵢⱼ. This scales the matrix uniformly.
Quick Reference
Note: Matrix multiplication (A×B) requires columns of A = rows of B. This calculator focuses on add, subtract, scalar, transpose, and inverse. For full matrix multiplication, use specialized tools.
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