2x2 Matrix Calculator

**2x2 Matrix:** A = [a₁₁ a₁₂] [a₂₁ a₂₂] **Determinant:** det(A) = a₁₁ × a₂₂ − a₁₂ × a₂₁ (Multiply main diagonal, subtract anti-diagonal product.) **Inverse:** A⁻¹ = (1/det(A)) × [a₂₂ −a₁₂] [−a₂₁ a₁₁] Exists only if det(A) ≠ 0. **Matrix multiplication (A × B):** If A = [a₁₁ a₁₂] B = [b₁₁ b₁₂] [a₂₁ a₂₂] [b₂₁ b₂₂] Then A × B = [a₁₁b₁₁ + a₁₂b₂₁ a₁₁b₁₂ + a₁₂b₂₂] [a₂₁b₁₁ + a₂₂b₂₁ a₂₁b₁₂ + a₂₂b₂₂] Row × column dot products. **Worked example:** A = [2 3] B = [5 6] [1 4] [7 8] det(A) = 2×4 − 3×1 = 5 A⁻¹ = (1/5) × [4 -3] = [0.8 -0.6] [-1 2] [-0.2 0.4] Verify: A × A⁻¹ should = I (identity). A × B = [2×5+3×7 2×6+3×8] = [31 38] [1×5+4×7 1×6+4×8] [33 38] **Properties of determinants:** | Property | Statement | |---|---| | Product | det(AB) = det(A) × det(B) | | Transpose | det(Aᵀ) = det(A) | | Inverse | det(A⁻¹) = 1/det(A) | | Identity | det(I) = 1 | | Zero column/row | det = 0 | | Row swap | det × -1 | | Row scaling by k | det × k | **Determinant interpretations:** - **Algebraic**: scalar invariant. - **Geometric**: signed area scaling of unit square. - **Sign**: orientation preservation (+) or reflection (-). - **Invertibility**: det ≠ 0 ⟺ matrix has inverse. **Identity matrix:** I = [1 0] [0 1] Any matrix × I = original matrix. **Matrix multiplication is NOT commutative:** A × B ≠ B × A (in general). Example: rotation × translation ≠ translation × rotation. Order matters! This makes matrix multiplication different from regular number multiplication. **But IS associative:** (A × B) × C = A × (B × C) So multi-matrix products can be grouped any way without changing result. **Transformations represented by 2x2 matrices:** | Matrix | Transformation | |---|---| | [1 0; 0 1] | Identity (no change) | | [k 0; 0 k] | Scaling by k | | [cos θ -sin θ; sin θ cos θ] | Rotation by θ | | [-1 0; 0 1] | Reflection over y-axis | | [1 0; 0 -1] | Reflection over x-axis | | [1 k; 0 1] | Horizontal shear by k | | [1 0; k 1] | Vertical shear by k | **Eigenvalues/Eigenvectors:** For matrix A, vector v is eigenvector with eigenvalue λ if: A × v = λ × v The matrix scales v by λ without changing direction. For 2x2 matrix: eigenvalues from characteristic equation: λ² − tr(A)λ + det(A) = 0 Where tr(A) = a₁₁ + a₂₂ (trace). **Common applications:** - **Computer graphics**: transformations (scale, rotate, translate). - **Physics**: rotations, Lorentz transformations. - **Quantum mechanics**: state vectors, operators. - **Computer vision**: image transformations. - **Robotics**: forward/inverse kinematics. - **Linear systems**: Ax = b equations. - **Statistics**: covariance matrices. - **Machine learning**: neural network layers. - **Economics**: input-output matrices. **Solving 2-variable linear systems:** System: ax + by = c, dx + ey = f. Matrix form: A × [x; y] = [c; f]. Solution: [x; y] = A⁻¹ × [c; f] (if A invertible). **Cramer's rule (alternative):** For Ax = b in 2 variables: x = det([b₁ a₁₂; b₂ a₂₂]) / det(A) y = det([a₁₁ b₁; a₂₁ b₂]) / det(A) Works for any 2x2 system with non-zero determinant. **Common 2x2 matrices:** | Matrix | Determinant | |---|---| | [1 0; 0 1] (identity) | 1 | | [2 0; 0 3] (diagonal) | 6 | | [0 -1; 1 0] (rotation 90°) | 1 | | [-1 0; 0 -1] (rotation 180°) | 1 | | [0 1; 1 0] (swap) | -1 | | [1 1; 0 1] (shear) | 1 | **Programming:** - **Python NumPy**: np.linalg.det(A), np.linalg.inv(A), np.matmul(A, B) or A @ B. - **MATLAB**: det(A), inv(A), A * B. - **C++ Eigen**: A.determinant(), A.inverse(), A * B. - **JavaScript math.js**: math.det(A), math.inv(A), math.multiply(A, B). **Software:** - **Wolfram Alpha**: full matrix operations. - **GeoGebra**: visual matrix transformations. - **NumPy/SciPy**: scientific computing standard. - **MATLAB**: traditional engineering. - **Mathematica**: symbolic linear algebra. **Higher-dimensional matrices:** For 3x3, 4x4, ..., nxn: same concepts but more complex calculations. - 3x3 determinant: cofactor expansion or rule of Sarrus. - 4x4+: usually use LU decomposition or other algorithms. For nxn: determinant has n! terms (5! = 120 for 5x5). Computers handle efficiently. **Connection to linear systems:** Ax = b - If det(A) ≠ 0: unique solution x = A⁻¹b. - If det(A) = 0: either no solution or infinitely many. Determinant is critical for solvability of linear systems. **Common pitfalls:** - **Singular matrices**: no inverse exists if determinant = 0. - **Non-commutativity**: A × B ≠ B × A in general. - **Wrong dimensions**: 2x2 × 2x2 = 2x2; 2x3 × 3x2 = 2x2 (inner dimensions must match). - **Row vs column indexing**: a₁₂ is row 1, column 2 (typically). - **Determinant for products**: det(AB) = det(A) × det(B), not (det(A) + det(B)) or other.