8  Determinants

A linear map T : \mathbb{R}^n \to \mathbb{R}^n transforms geometric objects—lines become lines, planes become planes, parallelograms become parallelograms. Distances and angles generally change, but certain properties persist. Among these is a remarkable invariant: the factor by which T scales volumes.

This chapter develops the determinant, a function \det : M_{n}(\mathbb{F}) \to \mathbb{F} that encodes this volume-scaling behavior. We characterize it axiomatically, derive computational methods, establish its fundamental properties, and explore its geometric content.

Remark. The determinant is also what we call an alternating multilinear k-form \det: \mathcal{V}^k \to \mathbb{F}, on an n-dimensional vector space V: a map that is linear in each slot, vanishes when two arguments agree, and changes sign when two arguments are swapped. More generally, alternating k-forms (“k-forms”) are k-linear alternating maps and play a central role in exterior algebra and differential geometry; viewing the determinant as an n-form explains its connection to oriented volume and is useful background for readers advancing to higher mathematics.

8.1 Motivation

Consider a linear map T : \mathbb{R}^2 \to \mathbb{R}^2 represented by matrix A \in M_{2}(\mathbb{R}). Apply T to the unit square [0,1]^2. The image is a parallelogram with vertices at 0, a_1, a_2, and a_1 + a_2, where a_1, a_2 are the columns of A.

The area of this parallelogram depends only on A—it measures how much T stretches or compresses area. If T doubles area, then every region’s area is doubled under T. If T collapses the plane onto a line (\ker(T) \neq \{0\}), then all areas become zero.

More generally, consider the unit sphere S^{n-1} \subset \mathbb{R}^n, the set of unit vectors. A linear map T : \mathbb{R}^n \to \mathbb{R}^n transforms S^{n-1} into an ellipsoid in \mathbb{R}^n. The volume of this ellipsoid, relative to the volume of S^{n-1}, is the volume scaling factor of T.

This scaling factor is intrinsic to T—it does not depend on which region we measure. If T scales all volumes by factor k, then for any measurable set E \subset \mathbb{R}^n with volume \text{vol}(E), the image T(E) has volume |k| \cdot \text{vol}(E). The absolute value accounts for orientation: if k < 0, the map reverses orientation.

Invariance under volume-preserving transformations. If S : \mathbb{R}^n \to \mathbb{R}^n is an isometry (distance-preserving map), then S preserves volumes. Rotations and reflections are isometries. If T scales volume by factor k, then S \circ T \circ S^{-1} also scales volume by factor k—conjugation by isometries does not change the scaling factor.

This invariance suggests that the volume scaling factor is intrinsic to the linear map T, not to any particular matrix representation. Two matrices representing the same linear map in different orthonormal bases have the same determinant. (The precise relationship between the determinant and the eigenvalues of T—which we study in Chapter 8—is that \det(T) equals the product of eigenvalues, counted with multiplicity, over an algebraically closed field.)

Why volume? Among all geometric quantities—length, angle, curvature—only volume has the multiplicativity property: if S scales volume by \alpha and T scales volume by \beta, then S \circ T scales volume by \alpha\beta. This multiplicativity, which we prove in Theorem 8.6, makes the determinant a powerful algebraic tool.

Projection and dimension collapse. If T is not injective, then \ker(T) \neq \{0\}. The entire kernel is collapsed to the origin, and the image \operatorname{im}(T) is a proper subspace of \mathbb{R}^n with \dim \operatorname{im}(T) < n by Theorem 4.5. Any n-dimensional region in \mathbb{R}^n is mapped into this lower-dimensional subspace.

Geometrically, the sphere S^{n-1} is flattened into an ellipsoid lying in a k-dimensional subspace where k = \operatorname{rank}(T) < n. An n-dimensional volume in \mathbb{R}^n cannot fit into a k-dimensional subspace without collapsing—the n-dimensional volume of T(S^{n-1}) is zero. This geometric fact corresponds to the algebraic fact that \det(T) = 0 when T is not invertible.

Observation. The volume-scaling factor is multiplicative: if S scales volume by factor \alpha and T scales volume by factor \beta, then S \circ T scales volume by factor \alpha\beta. This suggests the scaling factor should satisfy \det(AB) = \det(A)\det(B).

Observation. If the columns of A are linearly dependent, the image of the unit sphere lies in a lower-dimensional subspace, hence has zero n-dimensional volume. Conversely, if the columns are independent, T is invertible and the volume is nonzero. This suggests \det(A) = 0 \iff A is not invertible.

Observation. The scaling factor should depend continuously on the entries of A. Small perturbations to A produce small changes to the image of S^{n-1}, hence small changes in volume.

We seek a function with these properties. The challenge is to define it precisely and prove it exists uniquely.

8.2 Signed Volume and Orientation

In \mathbb{R}^n, the unit cube [0,1]^n has volume 1. A linear map T : \mathbb{R}^n \to \mathbb{R}^n sends this cube to a parallelepiped—the set \{ t_1 a_1 + \cdots + t_n a_n : 0 \le t_i \le 1 \} where a_1, \ldots, a_n are the columns of A = [T]_{\mathcal{E}} in the standard basis.

The volume of this parallelepiped measures how T scales n-dimensional volume. If \{a_1, \ldots, a_n\} are linearly dependent, the parallelepiped is degenerate (lies in a lower-dimensional subspace), and its n-dimensional volume is zero.

We introduce signed volume: the volume is assigned a sign depending on the orientation of the parallelepiped. The standard basis \{e_1, \ldots, e_n\} defines the positive orientation. If T preserves orientation, \det(T) > 0; if T reverses orientation, \det(T) < 0. If T collapses dimension, \det(T) = 0.

Orientation in \mathbb{R}^n. Two ordered bases of \mathbb{R}^n have the same orientation if one can be continuously deformed into the other through invertible linear maps without passing through a singular (non-invertible) map. Equivalently, two bases have the same orientation if the change-of-basis matrix between them has positive determinant.

In \mathbb{R}^2, the standard basis \{e_1, e_2\} defines counterclockwise orientation. Swapping the vectors to \{e_2, e_1\} reverses orientation to clockwise. In \mathbb{R}^3, the standard basis \{e_1, e_2, e_3\} defines right-handed orientation (the “right-hand rule” from physics).

Examples in \mathbb{R}^2.

  1. Identity: I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. The unit square is unchanged, so \det(I) = 1.

  2. Scaling: A = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} scales both directions by k, multiplying area by k^2. Thus \det(A) = k^2.

  3. Shear: A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} sends the unit square to a parallelogram with base 1 and height 1, area 1. Thus \det(A) = 1.

  4. Reflection: A = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} reflects across the y-axis, reversing orientation. The area is unchanged, so \det(A) = -1.

  5. Rotation: A = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} rotates by \theta, preserving area and orientation. Thus \det(A) = 1.

  6. Projection: A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} collapses the plane onto the x-axis. The image has zero 2-dimensional area, so \det(A) = 0.

These examples suggest that \det encodes both magnitude (how much volume changes) and sign (whether orientation is preserved).

Invariant subspaces. If T : \mathbb{R}^n \to \mathbb{R}^n leaves a subspace W \subseteq \mathbb{R}^n invariant—meaning T(w) \in W for all w \in W—then T decomposes into actions on W and its complementary subspace. When we develop eigenvalue theory in Chapter 8, we will see that eigenvectors span invariant subspaces of dimension 1, and the determinant is the product of the scaling factors (eigenvalues) along these directions.

For now, observe that if W is invariant and \dim W = k, we can choose a basis adapted to this decomposition, yielding a block-triangular matrix. The determinant of such a matrix factors into determinants of the diagonal blocks, reflecting the independent scalings on W and its complement.

8.3 Axiomatic Characterization

We define the determinant by specifying the properties it must satisfy, then prove such a function exists and is unique.

Definition 8.1 (Determinant (axiomatic)) A function \det : M_{n}(\mathbb{F}) \to \mathbb{F} is a determinant function if it satisfies:

  1. (Multilinearity in columns) For each j = 1, \ldots, n, the function is linear in the j-th column when the other columns are held fixed: \det(a_1, \ldots, a_{j-1}, ca_j + da_j', a_{j+1}, \ldots, a_n) = c\det(a_1, \ldots, a_j, \ldots, a_n) + d\det(a_1, \ldots, a_j', \ldots, a_n).

  2. (Alternating property) Swapping two columns negates the determinant: \det(a_1, \ldots, a_i, \ldots, a_j, \ldots, a_n) = -\det(a_1, \ldots, a_j, \ldots, a_i, \ldots, a_n).

  3. (Normalization) \det(I_n) = 1.

Here we write the matrix as a tuple of its columns: A = (a_1, \ldots, a_n) where a_j \in \mathbb{F}^n.

The multilinearity axiom reflects the fact that volume scales linearly with each edge of the parallelepiped. The alternating property encodes orientation. The normalization fixes the scale: the identity map preserves volume exactly.

Theorem 8.1 If \det satisfies axioms (1)–(3), then:

  1. If two columns of A are equal, then \det(A) = 0

  2. If the columns of A are linearly dependent, then \det(A) = 0

  3. Adding a multiple of one column to another does not change \det(A)

  4. Scaling a column by c multiplies \det(A) by c

Proof.

  1. If a_i = a_j for i \neq j, swapping columns i and j leaves A unchanged. By the alternating property, \det(A) = -\det(A), so 2\det(A) = 0. In a field of characteristic \neq 2 (which includes \mathbb{R} and \mathbb{C}), this gives \det(A) = 0.

  2. Suppose a_j = \sum_{i \neq j} c_i a_i. By multilinearity in column j, \det(a_1, \ldots, a_n) = \det\left(a_1, \ldots, \sum_{i \neq j} c_i a_i, \ldots, a_n\right) = \sum_{i \neq j} c_i \det(a_1, \ldots, a_i, \ldots, a_i, \ldots, a_n). Each term has two equal columns (column j equals column i), so by (a), each term vanishes.

  3. By multilinearity, \det(a_1, \ldots, a_i + ca_j, \ldots, a_j, \ldots, a_n) = \det(a_1, \ldots, a_i, \ldots, a_j, \ldots, a_n) + c\det(a_1, \ldots, a_j, \ldots, a_j, \ldots, a_n). The second term vanishes by (a).

  4. This is immediate from multilinearity. \square

These properties show that the determinant vanishes precisely when the columns are linearly dependent—consistent with the geometric interpretation that degenerate parallelepipeds have zero n-dimensional volume.

Remark. Property (c) means the determinant is unchanged by the row replacement operations used in Gaussian elimination (Chapter 6). This will provide an efficient computational method.

8.4 Permutations and the Leibniz Formula

To derive an explicit formula for the determinant, we expand using the multilinearity and alternating properties.

Definition 8.2 (Permutation) A permutation of \{1, 2, \ldots, n\} is a bijection \sigma : \{1, \ldots, n\} \to \{1, \ldots, n\}. The set of all permutations is denoted S_n, called the symmetric group of degree n.

Permutations can be composed: if \sigma, \tau \in S_n, then \sigma \circ \tau is the permutation applying \tau first, then \sigma. The identity permutation \text{id}(i) = i for all i is the neutral element. Every permutation \sigma has an inverse \sigma^{-1} with \sigma \circ \sigma^{-1} = \sigma^{-1} \circ \sigma = \text{id}.

A transposition is a permutation that swaps two elements and leaves all others fixed. Every permutation can be written as a composition of transpositions.

Definition 8.3 (Sign of a permutation) A permutation \sigma \in S_n can be written as a composition of transpositions. The sign of \sigma, denoted \operatorname{sgn}(\sigma) or \epsilon(\sigma), is +1 if \sigma is a composition of an even number of transpositions and -1 if odd.

The sign is well-defined: although a permutation can be written as a composition of transpositions in many ways, the parity (even or odd) is invariant. This can be proved by observing that the sign satisfies \operatorname{sgn}(\sigma \circ \tau) = \operatorname{sgn}(\sigma)\operatorname{sgn}(\tau) and that a single transposition has sign -1.

Examples. The identity has sign +1. Any single transposition has sign -1. In S_3, the cyclic permutation (1 \, 2 \, 3) sending 1 \to 2 \to 3 \to 1 can be written as (1 \, 2)(2 \, 3), two transpositions, so \operatorname{sgn}((1 \, 2 \, 3)) = +1.

Write each column of A in the standard basis: a_j = \sum_{i=1}^{n} a_{ij} e_i.

By repeated application of multilinearity, \det(A) = \det\left(\sum_{i_1} a_{i_1 1}e_{i_1}, \ldots, \sum_{i_n} a_{i_n n}e_{i_n}\right) = \sum_{i_1, \ldots, i_n} a_{i_1 1} \cdots a_{i_n n} \det(e_{i_1}, \ldots, e_{i_n}).

The sum is over all choices of indices i_1, \ldots, i_n \in \{1, \ldots, n\}. If any two indices are equal, the columns e_{i_1}, \ldots, e_{i_n} contain a repeat, so the determinant vanishes by Theorem 8.1(a). Thus only terms where (i_1, \ldots, i_n) is a permutation of (1, \ldots, n) contribute.

For \sigma \in S_n, the columns e_{\sigma(1)}, \ldots, e_{\sigma(n)} are obtained from e_1, \ldots, e_n by permuting. By repeated application of the alternating property, swapping adjacent columns, \det(e_{\sigma(1)}, \ldots, e_{\sigma(n)}) = \operatorname{sgn}(\sigma) \det(e_1, \ldots, e_n) = \operatorname{sgn}(\sigma).

Thus:

Theorem 8.2 (Leibniz formula) \det(A) = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \, a_{1\sigma(1)} a_{2\sigma(2)} \cdots a_{n\sigma(n)}.

Proof. Combining the expansion above with \det(e_{\sigma(1)}, \ldots, e_{\sigma(n)}) = \operatorname{sgn}(\sigma) and \det(I_n) = 1. \square

The sum has n! terms, one for each permutation. Each term is a product of n entries from A, one from each row and each column.

Example: n = 2. There are two permutations: \text{id} with sign +1 and the transposition (1 \, 2) with sign -1. Thus \det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc.

Example: n = 3. There are six permutations in S_3. The formula gives \det\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = aei + bfg + cdh - ceg - afh - bdi.

The Leibniz formula establishes existence and uniqueness of the determinant: it is the unique function satisfying axioms (1)–(3), and it is given explicitly by this sum.

8.5 Computational Methods

The Leibniz formula is impractical for large n—computing n! terms becomes infeasible. We develop more efficient methods based on row operations.

8.5.1 Row Operations and Determinants

Theorem 8.3 Let A \in M_{n}(\mathbb{F}).

  1. If B is obtained from A by swapping two rows, then \det(B) = -\det(A)

  2. If B is obtained from A by multiplying a row by scalar c, then \det(B) = c\det(A)

  3. If B is obtained from A by adding a multiple of one row to another, then \det(B) = \det(A)

Proof. We establish these using the Leibniz formula and properties of permutations.

  1. Swapping rows i and j in A produces matrix B with b_{k\ell} = a_{k\ell} for k \notin \{i,j\}, b_{i\ell} = a_{j\ell}, and b_{j\ell} = a_{i\ell}. In the Leibniz formula, \det(B) = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \, b_{1\sigma(1)} \cdots b_{n\sigma(n)}. Each term becomes a_{1\sigma(1)} \cdots a_{j\sigma(i)} \cdots a_{i\sigma(j)} \cdots a_{n\sigma(n)}, which equals a_{1\tau(1)} \cdots a_{n\tau(n)} where \tau is the permutation swapping \sigma(i) and \sigma(j). Since \tau = (i \, j) \circ \sigma, we have \operatorname{sgn}(\tau) = -\operatorname{sgn}(\sigma). The map \sigma \mapsto \tau is a bijection of S_n, so \det(B) = -\det(A).

  2. Multiplying row i by c means b_{i\ell} = ca_{i\ell} and b_{k\ell} = a_{k\ell} for k \neq i. Each term in the Leibniz formula for \det(B) contains exactly one factor from row i, namely b_{i\sigma(i)} = ca_{i\sigma(i)}. Thus each term is multiplied by c, giving \det(B) = c\det(A).

  3. Adding c times row j to row i (where i \neq j) gives b_{i\ell} = a_{i\ell} + ca_{j\ell} and b_{k\ell} = a_{k\ell} for k \neq i. By multilinearity (which we can verify holds for row operations using the Leibniz formula), \det(B) = \det(A) + c\det(A') where A' is the matrix with row i equal to row j and all other rows equal to those of A. Since A' has two equal rows, \det(A') = 0 by Theorem 8.1(a). \square

These properties allow us to compute determinants via Gaussian elimination.

8.5.2 Determinants of Triangular Matrices

Theorem 8.4 If A is upper or lower triangular, then \det(A) = a_{11} a_{22} \cdots a_{nn}, the product of the diagonal entries.

Proof. For upper triangular A, the Leibniz formula sum reduces to the single term corresponding to the identity permutation, since any other permutation would require choosing an entry a_{i\sigma(i)} with \sigma(i) < i, which is zero. Thus \det(A) = \operatorname{sgn}(\text{id}) \, a_{11} a_{22} \cdots a_{nn} = a_{11} a_{22} \cdots a_{nn}. The lower triangular case is similar. \square

8.5.3 Gaussian Elimination Algorithm

To compute \det(A):

  1. Reduce A to row echelon form U via row operations, tracking the effect on the determinant

  2. If a row swap is performed, multiply the running determinant by -1

  3. If a row is scaled by c, multiply the running determinant by c

  4. Row replacement operations do not change the determinant

  5. The final determinant is \det(U) = u_{11} u_{22} \cdots u_{nn}, adjusted by the accumulated signs and scaling factors

Since row echelon form is upper triangular, \det(U) is the product of the diagonal entries (the pivots, if no row scaling was used).

Corollary: A is invertible if and only if \det(A) \neq 0. This follows from Theorem 7.11: A is invertible if and only if it has n pivots (full rank), which occurs if and only if the diagonal entries of U are all nonzero, equivalently \det(U) \neq 0.

8.6 Cofactor Expansion

An alternative computational method expands the determinant recursively along a row or column.

Definition 8.4 (Minor and cofactor) Let A \in M_{n}(\mathbb{F}). The (i,j)-minor of A, denoted M_{ij}, is the determinant of the (n-1) \times (n-1) matrix obtained by deleting row i and column j from A.

The (i,j)-cofactor is C_{ij} = (-1)^{i+j} M_{ij}.

The factor (-1)^{i+j} accounts for the sign changes from moving the (i,j) entry to the top-left corner via row and column swaps.

Theorem 8.5 (Cofactor expansion) For any i \in \{1, \ldots, n\} (expansion along row i), \det(A) = \sum_{j=1}^{n} a_{ij} C_{ij}. For any j \in \{1, \ldots, n\} (expansion along column j), \det(A) = \sum_{i=1}^{n} a_{ij} C_{ij}.

Proof. We prove expansion along the first row; other cases follow by permuting rows/columns and tracking signs.

Write A = (a_1, \ldots, a_n) where a_j is the j-th column. Expanding a_1 = \sum_{i=1}^{n} a_{i1} e_i by multilinearity, \det(A) = \sum_{i=1}^{n} a_{i1} \det(e_i, a_2, \ldots, a_n).

For i > 1, swap row 1 with rows 2, \ldots, i successively (performing i-1 swaps), moving the nonzero entry of e_i to the top. This introduces a factor (-1)^{i-1}. The determinant of the resulting matrix has the form \det\begin{pmatrix} 1 & 0 & \cdots & 0 \\ * & & & \\ \vdots & & \tilde{A}_{i1} & \\ * & & & \end{pmatrix} where \tilde{A}_{i1} is the matrix obtained by deleting row i and column 1 from A. By multilinearity in the first column (or by direct calculation using the Leibniz formula), this determinant equals \det(\tilde{A}_{i1}) = M_{i1}.

Thus \det(A) = \sum_{i=1}^{n} a_{i1} (-1)^{i-1} M_{i1} = \sum_{i=1}^{n} a_{i1} C_{i1}, which is the expansion along column 1. Expansion along other columns follows by symmetry; expansion along rows uses \det(A^T) = \det(A), which we prove below. \square

Cofactor expansion reduces computing an n \times n determinant to computing n determinants of size (n-1) \times (n-1). For small n, this is practical. For large n, Gaussian elimination is more efficient (O(n^3) operations versus O(n!) for naive cofactor expansion).

8.7 Fundamental Properties

8.7.1 Multiplicativity

Theorem 8.6 For A, B \in M_{n}(\mathbb{F}), \det(AB) = \det(A) \det(B).

Proof. Define f : M_{n}(\mathbb{F}) \to \mathbb{F} by f(B) = \det(AB) / \det(A) for fixed A with \det(A) \neq 0. We verify f satisfies the determinant axioms.

For multilinearity: let B = (b_1, \ldots, b_n). Then AB = (Ab_1, \ldots, Ab_n). The determinant is multilinear in columns, so \det(A(b_1, \ldots, cb_j + db_j', \ldots, b_n)) = c\det(A(b_1, \ldots, b_j, \ldots, b_n)) + d\det(A(b_1, \ldots, b_j', \ldots, b_n)). Dividing by \det(A) shows f is multilinear in the columns of B.

For the alternating property: swapping columns j and k in B swaps columns j and k in AB, negating \det(AB). Thus f(B) changes sign.

For normalization: f(I_n) = \det(AI_n)/\det(A) = \det(A)/\det(A) = 1.

By uniqueness (Theorem 8.2), f = \det, so \det(AB)/\det(A) = \det(B), giving \det(AB) = \det(A)\det(B).

If \det(A) = 0, then A is not invertible, so \ker(A) \neq \{0\}. Hence A is not injective, so AB is not injective either (if AB were injective, then A restricted to \operatorname{im}(B) would be injective, but \ker(A) \neq \{0\} may not intersect \operatorname{im}(B)—more directly, there exists u \neq 0 with Au = 0; if B is surjective, write u = Bx to get ABx = 0 with the possibility that x \neq 0). The cleanest argument: AB not invertible follows because \operatorname{rank}(AB) \le \operatorname{rank}(A) < n. Thus \det(AB) = 0 = \det(A)\det(B). \square

This theorem formalizes the geometric observation: if A scales volume by \alpha and B scales volume by \beta, then AB scales volume by \alpha\beta.

Corollary: If A is invertible, then \det(A^{-1}) = 1/\det(A). This follows from \det(AA^{-1}) = \det(I_n) = 1.

8.7.2 Transpose

Theorem 8.7 For any A \in M_{n}(\mathbb{F}), \det(A^T) = \det(A).

Proof. The Leibniz formula for A is \det(A) = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \, a_{1\sigma(1)} \cdots a_{n\sigma(n)}. For A^T, the (i,j) entry is a_{ji}, so \det(A^T) = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \, a_{\sigma(1)1} a_{\sigma(2)2} \cdots a_{\sigma(n)n}. Reindex by \tau = \sigma^{-1}. Then \sigma(i) = j \iff \tau(j) = i, so the product becomes a_{\sigma(1)1} \cdots a_{\sigma(n)n} = a_{1\tau(1)} \cdots a_{n\tau(n)}. Since \operatorname{sgn}(\sigma) = \operatorname{sgn}(\tau) (the inverse of a permutation has the same sign), and the map \sigma \mapsto \tau is a bijection of S_n, \det(A^T) = \sum_{\tau \in S_n} \operatorname{sgn}(\tau) \, a_{1\tau(1)} \cdots a_{n\tau(n)} = \det(A). \quad \square

This theorem justifies applying row operations in place of column operations when computing determinants.

8.8 Geometric Applications

8.8.1 Volume of Parallelepipeds

The determinant directly computes the signed volume of the parallelepiped spanned by the columns of A.

Theorem 8.8 Let v_1, \ldots, v_n \in \mathbb{R}^n and A = (v_1 \, \cdots \, v_n). The n-dimensional volume of the parallelepiped \mathcal{P} = \{ t_1 v_1 + \cdots + t_n v_n : 0 \le t_i \le 1 \} is |\det(A)|.

This is the rigorous formulation of our initial motivation. The absolute value accounts for orientation—volume is always nonnegative, but the determinant may be negative.

Example. In \mathbb{R}^2, vectors v_1 = (a,c) and v_2 = (b,d) span a parallelogram with area |ad - bc|, which is |\det(A)| for A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}.

8.8.2 Change of Variables in Integration

In multivariable calculus, change of variables in integration uses the determinant of the Jacobian matrix. If T : \mathbb{R}^n \to \mathbb{R}^n is a differentiable map with Jacobian matrix J_T(x), then \int_{T(E)} f(y) \, dy = \int_E f(T(x)) |\det(J_T(x))| \, dx.

For linear maps T(x) = Ax, the Jacobian is constant: J_T(x) = A. The change of variables formula becomes \int_{T(E)} f(y) \, dy = |\det(A)| \int_E f(Ax) \, dx.

The factor |\det(A)| is the volume scaling factor we motivated at the chapter’s outset.

8.9 Block Matrices and Determinants

When a matrix has block structure, the determinant often simplifies.

Theorem 8.9 If A \in M_{n}(\mathbb{F}) has block-triangular form A = \begin{pmatrix} B & C \\ 0 & D \end{pmatrix} where B is k \times k and D is (n-k) \times (n-k), then \det(A) = \det(B) \det(D).

Proof. We can reduce this to the scalar case by induction on n, or observe directly using the Leibniz formula: any permutation \sigma contributing to \det(A) must satisfy \sigma(i) \le k for i \le k (else we’d select from the zero block). This forces \sigma to permute \{1, \ldots, k\} among themselves and \{k+1, \ldots, n\} among themselves. The product factorizes into contributions from B and D, and the sign of \sigma is the product of signs from the two blocks. \square

This theorem confirms our earlier observation about invariant subspaces: if a matrix decomposes into independent actions on complementary subspaces, the determinant is the product of determinants on each piece.

8.10 Closing Remarks

The determinant packages several fundamental properties of a linear map into a single scalar: it detects invertibility, measures volume scaling, encodes orientation, and satisfies multiplicativity. Its axiomatic characterization—multilinearity, alternating property, normalization—uniquely determines an explicit formula (Leibniz), from which computational methods (row reduction, cofactor expansion) follow.

The determinant is basis-independent in a crucial sense: similar matrices have equal determinants. If B = P^{-1}AP, then \det(B) = \det(P^{-1})\det(A)\det(P) = \det(A). This reflects the fact that the volume scaling factor of a linear map does not depend on coordinate choice.

In the next chapter, we develop eigenvalue theory.