Chapter 03: Determinants¶

Prerequisites: Matrix Algebra (Ch02)

Chapter Outline: Definition of Determinant (Permutations & Inversions) → 2x2 and 3x3 Intuition → Key Properties (Effect of Row/Column Operations) → Minors and Cofactors → Laplace Expansion Theorem → Determinants of Products and Inverses → Adjoint Matrix $A^*$ & Inverse Formula → Cramer's Rule → Geometric Interpretation (Volume Scaling Factor)

Extension: The determinant is the unique scalar indicator of matrix invertibility and serves as the polynomial engine for studying Eigenvalues (Ch06).

The determinant is a function that maps a square matrix to a scalar value. Although solving systems via determinants is computationally inefficient, they hold an irreplaceable position in theoretical analysis, area/volume calculation, and eigenvalue determination. This chapter begins with the algebraic definition using permutations and ultimately reveals its geometric essence.

03.1 Definition of the Determinant¶

Definition 03.1 (Permutations and Inversions)

The number of inversions $\tau(\sigma)$ in a permutation of $n$ numbers is the count of pairs where a larger number precedes a smaller one. The sign of a permutation is defined as $\operatorname{sgn}(\sigma) = (-1)^{\tau(\sigma)}$.

Definition 03.2 (Leibniz Formula)

The determinant of an $n \times n$ matrix $A$, denoted $\det(A)$ or $|A|$, is defined as: $$\det(A) = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) a_{1,\sigma(1)} a_{2,\sigma(2)} \cdots a_{n,\sigma(n)}$$

03.2 Properties of Determinants¶

Theorem 03.1 (Core Properties)

Transpose Invariance: $\det(A^T) = \det(A)$.
Multiplicative Property: $\det(AB) = \det(A) \det(B)$.
Row Operation Effects:
Swapping two rows flips the sign.
Scaling a row by $k$ scales the determinant by $k$.
Adding a multiple of one row to another does not change the determinant.
Invertibility: $A$ is invertible $\iff \det(A) \neq 0$.

03.3 Minors and Laplace Expansion¶

Definition 03.3 (Minors and Cofactors)

The determinant of the submatrix formed by deleting the $i$-th row and $j$-th column of $A$ is the minor $M_{ij}$. The cofactor is defined as $C_{ij} = (-1)^{i+j} M_{ij}$.

Theorem 03.2 (Laplace Expansion)

The determinant equals the sum of the products of elements of any row (or column) and their corresponding cofactors: $$\det(A) = \sum_{j=1}^n a_{ij} C_{ij} \quad (\text{for a fixed } i)$$

03.4 The Adjoint Matrix and Cramer's Rule¶

Definition 03.4 (Adjoint Matrix $A^*$)

The transpose of the matrix of cofactors is called the adjoint (or adjugate) matrix of $A$: $$(A^*)_{ij} = C_{ji}$$ It satisfies the identity: $AA^* = A^*A = \det(A)I$. If $\det(A) \neq 0$, then $A^{-1} = \frac{1}{\det(A)} A^*$.

Application: Cramer's Rule

For a system $Ax = b$, if $\det(A) \neq 0$, then $x_j = \frac{\det(A_j)}{\det(A)}$, where $A_j$ is obtained by replacing the $j$-th column of $A$ with $b$.

Exercises¶

Solution

Chapter Summary¶

The determinant is the "volume tag" of a square matrix:

****: Whether the determinant is zero is the ultimate divide between invertible and singular matrices, and between systems with or without unique solutions.

****: Laplace expansion recursion connects high-order determinants to lower-order submatrices, while the multiplicative property demonstrates the accumulation of volume after the composition of linear operators.

****: The adjoint matrix not only provides an explicit formula for inversion but also deeply reveals the algebraic relationship between a matrix and its internal sub-blocks.