Determinants - Linear Algebra

Definition and Basic Concepts

Square Matrices

Determinants defined only for square matrices (n×n). Output: scalar value summarizing matrix properties.

Recursive Definition

For 1×1 matrix A = [a], det(A) = a. For n>1, det(A) defined via expansion by minors or cofactors.

Notation

Denoted as det(A) or |A|. Compact and widely accepted in literature.

Initial Examples

2×2: det([[a, b], [c, d]]) = ad - bc. 3×3: sum of products of diagonals minus sum of anti-diagonals.

Properties of Determinants

Linearity

Determinant is multilinear in rows (or columns): linear function in each row with others fixed.

Alternating Property

Swapping two rows (or columns) multiplies determinant by -1.

Normalization

Determinant of identity matrix I_n is 1.

Multiplicativity

det(AB) = det(A) * det(B) for square matrices A, B.

Effect of Scaling

Multiplying a row by scalar k multiplies determinant by k.

Calculation Methods

Direct Expansion

Using Laplace expansion, computationally expensive for large n.

Row Reduction

Transform matrix to upper-triangular form; determinant is product of diagonal entries adjusted by row operation effects.

Permutation Definition

Sum over all permutations σ ∈ S_n: det(A) = Σ (sign(σ) ∏ a_{i,σ(i)}), factorial complexity.

Leveraging Sparsity

Zero elements reduce computational complexity in expansions and elimination.

Minors and Cofactors

Minor

Minor M_{ij} of element a_{ij}: determinant of (n-1)×(n-1) matrix after removing i-th row and j-th column.

Cofactor

Cofactor C_{ij} = (-1)^{i+j} M_{ij}; sign alternates in checkerboard pattern.

Use in Expansion

Determinant expressed as sum of a_{ij} * C_{ij} along any row or column.

Matrix of Cofactors

Matrix formed by all cofactors; used in adjugate and inverse calculations.

Laplace Expansion

Definition

Expands determinant along any row or column using cofactors.

Formula

det(A) = Σ_{j=1}^n a_{ij} C_{ij} (expansion along i-th row)det(A) = Σ_{i=1}^n a_{ij} C_{ij} (expansion along j-th column)

Choice of Row/Column

Optimal to choose row/column with most zeros to minimize computation.

Recursive Nature

Expansion reduces determinant of n×n matrix to determinants of (n-1)×(n-1) minors.

Effect of Row Operations

Swapping Rows

Exchanges sign of determinant: det(B) = -det(A) if B formed by swapping two rows of A.

Scaling a Row

Multiplying a row by scalar k multiplies determinant by k.

Adding Multiple of One Row to Another

Does not change determinant value.

Use in Computation

Row operations simplify determinant calculation via triangularization.

Determinant of Product and Inverse

Multiplicative Property

det(AB) = det(A) * det(B) for square matrices A, B.

Determinant of Inverse

If A invertible, det(A^{-1}) = 1 / det(A).

Determinant of Transpose

det(A^T) = det(A).

Consequences

Multiplicativity used in proofs and matrix factorization properties.

Geometric Interpretation

Volume Scaling Factor

Determinant gives scaling factor of n-dimensional volume under linear transformation represented by A.

Orientation

Sign of determinant indicates orientation preservation (+) or reversal (-).

Examples

2×2 matrix determinant: area scaling of parallelogram spanned by column vectors.

Zero Determinant

Indicates volume collapse to lower dimension; transformation is singular.

Applications of Determinants

Solving Linear Systems

Cramer's Rule: solution to linear system using ratio of determinants.

Matrix Inversion

Inverse computed via adjugate matrix and determinant.

Eigenvalues and Characteristic Polynomial

Determinant of (A - λI) used to find eigenvalues (det = 0).

Testing Linear Independence

Nonzero determinant indicates linearly independent vectors.

Change of Variables in Integrals

Jacobian determinant transforms integration variables.

Singularity and Invertibility

Singular Matrices

Determinant zero: matrix singular, no inverse.

Invertible Matrices

Nonzero determinant: matrix invertible.

Rank and Determinant

Full rank n implies nonzero determinant.

Practical Implications

Singularity indicates dependent system, no unique solutions.

Determinants of Special Matrices

Diagonal Matrices

Determinant equals product of diagonal entries.

Triangular Matrices

Determinant equals product of diagonal entries (upper or lower triangular).

Orthogonal Matrices

Determinant ±1; reflects rotation (1) or reflection (-1).

Permutation Matrices

Determinant equals sign of permutation (+1 or -1).

Matrix Type	Determinant Formula
Diagonal	Product of diagonal entries ∏ a_{ii}
Triangular	Product of diagonal entries ∏ a_{ii}
Orthogonal	±1
Permutation	Sign of permutation (+1 or -1)

Computational Algorithms

LU Decomposition

Factor A into L (lower triangular) and U (upper triangular); det(A) = det(L)*det(U), product of diagonal entries.

QR Decomposition

Decompose A = QR; det(A) = det(Q) * det(R); det(Q) = ±1, det(R) product of diagonals.

Recursive Algorithms

Recursive expansion by minors feasible for small matrices; factorial complexity.

Numerical Stability

Row reduction and decomposition methods preferred for numerical accuracy.

Algorithm: Determinant via LU DecompositionInput: Square matrix A (n×n)Output: det(A)1. Compute LU factorization: A = L * U (with pivoting if necessary)2. det(L) = product of diagonal entries of L (typically 1 if unit lower triangular)3. det(U) = product of diagonal entries of U4. det(A) = det(L) * det(U)5. Adjust sign if row exchanges (pivoting) performed during factorization

References

G. Strang, Introduction to Linear Algebra, Wellesley-Cambridge Press, Vol. 5, 2016, pp. 45-78.
K. Hoffman and R. Kunze, Linear Algebra, Prentice-Hall, 2nd ed., 1971, pp. 120-150.
R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1985, pp. 35-60.
D.C. Lay, Linear Algebra and Its Applications, Pearson, 4th ed., 2011, pp. 85-110.
S. Axler, Linear Algebra Done Right, Springer, 3rd ed., 2015, pp. 90-115.