2×2 and 3×3 Matrix Calculator

Matrices Guide

Matrix Operations

A matrix is a rectangular array of numbers. For a 2×2 matrix A = [[a,b],[c,d]]: determinant det(A) = ad − bc. Inverse A⁻¹ = (1/det(A)) × [[d,−b],[−c,a]] (only exists if det(A) ≠ 0). Transpose Aᵀ: swap rows and columns — row 1 becomes column 1, etc. Matrix multiplication AB: element (i,j) = row i of A dotted with column j of B. Note: AB ≠ BA in general (matrix multiplication is not commutative).

The Determinant and Geometric Meaning

For a 2×2 matrix, the determinant equals the signed area of the parallelogram formed by the column vectors. det = 0 means the matrix is singular (rows/columns are linearly dependent, the parallelogram has collapsed to a line or point). The matrix cannot be inverted. For a 3×3 matrix: det = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁) — expansion along the first row. A 3×3 determinant gives the signed volume of the parallelepiped formed by the three column vectors.

Solving Linear Systems with Matrices

A system of linear equations can be written as Ax = b. If det(A) ≠ 0, the unique solution is x = A⁻¹b. For 2 equations: 2x + y = 5, x + 3y = 7. Matrix form: A = [[2,1],[1,3]], b = [[5],[7]]. det(A) = 6−1 = 5. A⁻¹ = (1/5)[[3,−1],[−1,2]]. x = A⁻¹b = (1/5)[[3×5 + (−1)×7],[( −1)×5 + 2×7]] = (1/5)[[8],[9]] = [[1.6],[1.8]]. Check: 2(1.6)+1.8 = 5 ✓.

Applications of Matrices

Matrices appear throughout applied mathematics and engineering. Computer graphics: transformation matrices rotate, scale, and translate 3D objects. Quantum mechanics: quantum states are represented as vectors, operators as matrices. Statistics: variance-covariance matrices in multivariate analysis. Network analysis: adjacency matrices represent connections. Machine learning: neural networks process data through sequential matrix multiplications. Economics: Leontief input-output model uses matric