Determinant
The determinant is a scalar value computed from a square matrix that encodes whether the linear transformation represented by the matrix is invertible, scaling the volume by a factor equal to the absolute value of the determinant and changing orientation if the determinant is negative. For a 2×2 matrix, det(A) = ad − bc; for larger matrices, it is computed recursively via cofactor expansion or row reduction. The determinant is zero if and only if the matrix is singular (non-invertible), making it a critical tool in solving linear systems, computing eigenvalues, and computing cross products in geometry.
Key Formula
det(A) = sum over j of a_ij * C_ij, where C_ij = (-1)^(i+j) * M_ij
LaTeX: \det(A) = \sum_{j=1}^{n} a_{ij} \, C_{ij}, \quad C_{ij} = (-1)^{i+j} M_{ij}
| Symbol | Meaning | Unit |
|---|---|---|
| a_{ij} | Entry in row i, column j | — |
| C_{ij} | Cofactor of entry a_ij | — |
| M_{ij} | Minor — determinant of submatrix after deleting row i and column j | — |
Worked Example
Problem
Find the determinant of A = [[3, 8], [4, 6]].
Solution
Step 1: Apply the 2×2 formula: det(A) = ad − bc. Step 2: a = 3, b = 8, c = 4, d = 6. Step 3: det(A) = (3)(6) − (8)(4) = 18 − 32 = −14.
Answer
det(A) = −14. The matrix is invertible (det ≠ 0); the transformation reverses orientation (det < 0).
Determinant Properties Summary
| Property | Statement | Implication |
|---|---|---|
| Invertibility | det(A) ≠ 0 | Matrix has an inverse |
| Singular matrix | det(A) = 0 | Columns/rows are linearly dependent |
| Product rule | det(AB) = det(A)·det(B) | Determinant is multiplicative |
| Transpose | det(Aᵀ) = det(A) | Same determinant as original |
| Row swap | Swapping two rows negates det | Sign changes with each swap |
| Scalar multiple | det(kA) = kⁿ det(A) | Scaling all entries scales det by kⁿ |
Interactive Tools
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Related Terms
Matrix (Linear Algebra)
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns, where an m×n matrix has m rows and n columns. Matrices represent linear transformations between vector spaces, encode systems of linear equations, and serve as the primary computational tool in linear algebra. They are indispensable across engineering, physics, computer science, statistics, and machine learning, where they model data, rotations, reflections, and the weights of neural networks.
Eigenvalue
An eigenvalue of a square matrix A is a scalar λ such that there exists a non-zero vector v (its eigenvector) satisfying Av = λv, meaning the matrix only scales the vector without changing its direction. Eigenvalues are found by solving the characteristic equation det(A − λI) = 0, and they encode fundamental properties of the linear transformation including stability, principal stresses, and oscillation frequencies. Applications span quantum mechanics (energy levels), structural engineering (natural frequencies), principal component analysis (data variance), and PageRank (web link importance).
Matrix Rank
The rank of a matrix is the dimension of the vector space spanned by its columns (column rank), which always equals the dimension spanned by its rows (row rank), giving a single well-defined measure of the "information content" or degrees of freedom in the matrix. A matrix A of size m×n has rank at most min(m, n); if the rank equals min(m, n) the matrix is called full rank. The rank determines whether a linear system Ax = b has a unique solution (full rank square matrix), infinitely many solutions, or no solution, making it central to the theory of linear equations, least-squares fitting, and dimensionality reduction.
From Latin "determinare" (to limit, to define). The term was introduced by Carl Friedrich Gauss in 1801 in "Disquisitiones Arithmeticae", though the concept was studied earlier by Leibniz (1693) and Cramer (1750).