LA-C8

Available

Course 8: Determinants

The determinant is a fundamental scalar-valued function on square matrices that measures how a matrix scales volume and determines invertibility. This course covers the definition, properties, computation methods, and applications of determinants.

12-15 hours Core Level 10 Objectives

Learning Objectives

State and apply the axiomatic definition of determinants (multilinear, alternating, normalized).
Master the permutation (Leibniz) formula with inversion numbers.
Prove and apply multiplicativity: det(AB) = det(A)det(B).
Understand the connection between determinant and invertibility.
Compute determinants using row reduction and cofactor expansion.
Apply Laplace expansion along any row or column strategically.
Understand the adjugate matrix and the inverse formula A⁻¹ = adj(A)/det(A).
Apply Cramer's rule to solve linear systems.
Recognize special determinants (triangular, Vandermonde, block).
Understand the geometric interpretation as signed volume.

Prerequisites

LA-C7: Matrix Inverses, Elementary Matrices & Dual Spaces
Matrix operations and elementary matrices
Permutations and basic combinatorics
Gaussian elimination
Basic properties of linear maps

Historical Context

Determinants were first studied by Gottfried Wilhelm Leibniz (1646–1716) and Seki Takakazu (1642–1708) independently in the late 17th century. The modern axiomatic definition was developed by Karl Weierstrass and others in the 19th century. Pierre-Simon Laplace (1749–1827) developed the cofactor expansion method, while Gabriel Cramer (1704–1752) gave the rule for solving linear systems. The geometric interpretation as signed volume was recognized early, and determinants remain central to linear algebra, differential geometry, and many areas of mathematics.

1. Definition of Determinant

The determinant can be defined in multiple equivalent ways: axiomatically (multilinear, alternating, normalized), via permutations (Leibniz formula), or recursively via cofactor expansion.

Definition 1.1: Axiomatic Definition

The determinant is the unique function $\det: M_n(F) \to F$ satisfying:

Multilinearity: Linear in each row (and column)
Alternating: Swapping two rows (or columns) negates the determinant
Normalization: $\det(I) = 1$

Definition 1.2: Permutation (Leibniz) Formula

For an $n \times n$ matrix $A$ :

\det(A) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{i=1}^n a_{i,\sigma(i)}

where $S_n$ is the symmetric group and $\text{sgn}(\sigma) = (-1)^{\text{inv}(\sigma)}$ is the sign of permutation $\sigma$ .

Example 1.1: 2×2 Determinant

For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ :

\det(A) = ad - bc

This comes from two permutations: identity (gives $+ad$ ) and swap (gives $-bc$ ).

Remark 1.1: Geometric Interpretation

For $2 \times 2$ , $|\det(A)|$ is the area of the parallelogram spanned by the columns. For $3 \times 3$ , it's the signed volume of the parallelepiped. The sign indicates orientation.

Definition 1.3: Recursive Definition via Minors

For $n > 1$ , expanding along the first row:

\det(A) = \sum_{j=1}^n (-1)^{1+j} a_{1j} \det(M_{1j})

where $M_{1j}$ is the $(n-1) \times (n-1)$ matrix obtained by deleting row 1 and column $j$ .

Example 1.2: 3×3 Determinant

For $A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$ :

\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

This is the "Sarrus rule" for $3 \times 3$ matrices (valid only for $3 \times 3$ ).

2. Properties of Determinants

Determinants have beautiful algebraic properties that make them powerful tools for understanding matrices and linear maps.

Theorem 2.1: Multiplicativity

$\det(AB) = \det(A) \det(B)$ for any $n \times n$ matrices $A, B$ .

Theorem 2.2: Transpose Property

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

Theorem 2.3: Invertibility Criterion

An $n \times n$ matrix $A$ is invertible if and only if $\det(A) \neq 0$ .

Corollary 2.1: Inverse Determinant

If $A$ is invertible, then $\det(A^{-1}) = 1/\det(A)$ .

Theorem 2.4: Row Operations

Swapping two rows: $\det \to -\det$
Scaling a row by $c$ : $\det \to c \cdot \det$
Adding a multiple of one row to another: $\det$ unchanged

Theorem 2.5: Triangular Matrices

For triangular (upper or lower) matrices, $\det = \prod_{i=1}^n a_{ii}$ (product of diagonal entries).

Example 2.1: Scalar Multiple

For an $n \times n$ matrix $A$ and scalar $c$ :

\det(cA) = c^n \det(A)

Each of $n$ rows gets multiplied by $c$ , so the determinant scales by $c^n$ .

Theorem 2.6: Block Matrix Determinant

For block matrices:

If $A = \begin{pmatrix} B & C \\ 0 & D \end{pmatrix}$ with $B, D$ square, then $\det(A) = \det(B) \det(D)$
Similarly for block lower triangular matrices

Corollary 2.2: Determinant of Similar Matrices

If $B = P^{-1} A P$ , then $\det(B) = \det(A)$ . Similar matrices have the same determinant.

3. Computation Methods

There are several methods for computing determinants, each with different computational complexity and practical advantages.

Definition 3.1: Row Reduction Method

Row reduce $A$ to upper triangular form $U$ , tracking:

Each row swap multiplies det by $-1$
Each row scaling by $c$ multiplies det by $c$
Row additions don't change det

Then $\det(A) = \det(U) / (\text{product of scaling factors}) \times (-1)^{\text{number of swaps}}$ .

Remark 3.1: Complexity

Row reduction is $O(n^3)$ , making it the preferred method for large matrices.

Definition 3.2: Cofactor Expansion (Laplace)

For any row $i$ or column $j$ :

\det(A) = \sum_{j=1}^n a_{ij} A_{ij} = \sum_{i=1}^n a_{ij} A_{ij}

where $A_{ij} = (-1)^{i+j} M_{ij}$ is the cofactor and $M_{ij}$ is the minor (determinant after deleting row $i$ and column $j$ ).

Remark 3.2: Strategic Choice

Choose a row/column with many zeros to minimize computation. Cofactor expansion is $O(n!)$ in general but can be efficient for sparse matrices.

Example 3.1: Vandermonde Determinant

For $V(x_1, \ldots, x_n) = \det\begin{pmatrix} 1 & x_1 & x_1^2 & \cdots \\ 1 & x_2 & x_2^2 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{pmatrix}$ :

\det(V) = \prod_{1 \leq i < j \leq n} (x_j - x_i)

This is crucial in polynomial interpolation theory.

Example 3.2: Determinant via LU Decomposition

If $A = LU$ where $L$ is lower triangular and $U$ is upper triangular, then:

\det(A) = \det(L) \det(U) = (\prod l_{ii}) (\prod u_{ii})

This is efficient when $A$ can be factored as $LU$ .

Remark 3.3: Computational Strategies

For small matrices (≤3): use direct formula
For sparse matrices: use cofactor expansion along rows/columns with many zeros
For general matrices: use row reduction (most efficient)
For structured matrices: use specialized formulas (Vandermonde, block, etc.)

4. Laplace Expansion and Adjugate

Laplace expansion provides a recursive method for computing determinants, and the adjugate matrix gives an explicit formula for the inverse.

Theorem 4.1: Laplace Expansion

For any row $i$ :

\det(A) = \sum_{j=1}^n a_{ij} A_{ij}

Similarly for any column $j$ . The expansion is valid along any row or column.

Theorem 4.2: Alien Cofactor Theorem

If $i \neq k$ , then $\sum_{j=1}^n a_{ij} A_{kj} = 0$ .

This is like computing the determinant of a matrix with two identical rows.

Definition 4.1: Adjugate Matrix

The adjugate (or classical adjoint) of $A$ is:

[\text{adj}(A)]_{ij} = A_{ji}

where $A_{ji}$ is the cofactor. That is, $\text{adj}(A)$ is the transpose of the cofactor matrix.

Theorem 4.3: Fundamental Adjugate Identity

$A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = \det(A) \cdot I$ .

Corollary 4.1: Inverse Formula

If $\det(A) \neq 0$ , then $A^{-1} = \frac{\text{adj}(A)}{\det(A)}$ .

Example 4.1: 2×2 Adjugate

For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ :

\text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Note: swap diagonal, negate off-diagonal.

Definition 4.2: Cramer's Rule

For the system $Ax = b$ with $\det(A) \neq 0$ :

x_i = \frac{\det(A_i)}{\det(A)}

where $A_i$ is $A$ with column $i$ replaced by $b$ .

Remark 4.1: Computational Note

Cramer's rule is elegant but computationally expensive ( $O(n \cdot n!)$ ). For practical computation, use Gaussian elimination ( $O(n^3)$ ).

Example 4.2: Cramer's Rule Example

Solve $\begin{cases} 2x + y = 5 \\ x + 3y = 7 \end{cases}$ using Cramer's rule.

$A = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}$ , $b = \begin{pmatrix} 5 \\ 7 \end{pmatrix}$

$\det(A) = 5$ , $\det(A_1) = \det\begin{pmatrix} 5 & 1 \\ 7 & 3 \end{pmatrix} = 8$ , $\det(A_2) = \det\begin{pmatrix} 2 & 5 \\ 1 & 7 \end{pmatrix} = 9$

So $x = 8/5$ , $y = 9/5$ .

Theorem 4.4: Adjugate and Inverse

The adjugate provides an explicit formula for the inverse:

A^{-1} = \frac{1}{\det(A)} \text{adj}(A)

This is particularly useful for symbolic computation and theoretical analysis.

5. Advanced Determinant Techniques

Advanced techniques for computing and manipulating determinants, including special matrix types and computational tricks.

Theorem 5.1: Determinant of Block Diagonal

If $A = \text{diag}(A_1, A_2, \ldots, A_k)$ is block diagonal, then:

\det(A) = \prod_{i=1}^k \det(A_i)

Theorem 5.2: Schur Complement Formula

For block matrix $A = \begin{pmatrix} B & C \\ D & E \end{pmatrix}$ with $B$ invertible:

\det(A) = \det(B) \det(E - DB^{-1}C)

The matrix $E - DB^{-1}C$ is called the Schur complement of $B$ in $A$ .

Example 5.1: Computing Large Determinants

For a $10 \times 10$ matrix with block structure, use Schur complement to reduce to smaller determinants.

Definition 5.1: Wronskian

For functions $f_1, \ldots, f_n$ , the Wronskian is:

W(f_1, \ldots, f_n) = \det\begin{pmatrix} f_1 & f_2 & \cdots & f_n \\ f_1' & f_2' & \cdots & f_n' \\ \vdots & \vdots & \ddots & \vdots \\ f_1^{(n-1)} & f_2^{(n-1)} & \cdots & f_n^{(n-1)} \end{pmatrix}

If $W \neq 0$ at some point, the functions are linearly independent.

Theorem 5.3: Cauchy-Binet Formula

For $A \in M_{m \times n}(F)$ and $B \in M_{n \times m}(F)$ with $m \leq n$ :

\det(AB) = \sum_{S} \det(A_S) \det(B_S)

where the sum is over all $m$ -element subsets $S$ of $\{1, \ldots, n\}$ , and $A_S, B_S$ are the corresponding submatrices.

Example 5.2: Determinant of Product of Non-Square Matrices

For $A \in M_{2 \times 3}$ and $B \in M_{3 \times 2}$ , $\det(AB)$ can be computed using Cauchy-Binet formula.

6. Determinants and Geometry

Determinants have deep geometric interpretations: they measure volumes, areas, and orientation. This connection between algebra and geometry is fundamental to linear algebra.

Theorem 6.1: Volume Interpretation

For $A \in M_n(\mathbb{R})$ , $|\det(A)|$ is the $n$ -dimensional volume of the parallelepiped spanned by the columns (or rows) of $A$ .

Example 6.1: Area in ℝ²

For $A = \begin{pmatrix} a & c \\ b & d \end{pmatrix}$ , the columns are $(a, b)$ and $(c, d)$ .

The area of the parallelogram is $|ad - bc| = |\det(A)|$ .

Example 6.2: Volume in ℝ³

For $3 \times 3$ matrix $A$ , $|\det(A)|$ is the volume of the parallelepiped spanned by the three column vectors.

If $\det(A) = 0$ , the vectors are coplanar (volume is zero).

Definition 6.1: Orientation

The sign of $\det(A)$ indicates orientation:

$\det(A) > 0$ : preserves orientation (right-handed)
$\det(A) < 0$ : reverses orientation (left-handed)
$\det(A) = 0$ : collapses to lower dimension

Theorem 6.2: Change of Variables

For a linear transformation $T: \mathbb{R}^n \to \mathbb{R}^n$ with matrix $A$ , if $S$ is a region with volume $V$ , then $T(S)$ has volume $|\det(A)| \cdot V$ .

Example 6.3: Scaling and Rotation

Scaling by factor $k$ : $\det = k^n$ (volume scales by $k^n$ )
Rotation: $\det = 1$ (preserves volume and orientation)
Reflection: $\det = -1$ (preserves volume, reverses orientation)

Theorem 6.3: Cross Product and Determinant

In $\mathbb{R}^3$ , the cross product $u \times v$ can be computed as:

u \times v = \det\begin{pmatrix} e_1 & e_2 & e_3 \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{pmatrix}

The magnitude $|u \times v|$ equals the area of the parallelogram spanned by $u$ and $v$ .

Example 6.4: Triangle Area

For a triangle with vertices $P, Q, R$ in $\mathbb{R}^2$ , the area is:

\text{Area} = \frac{1}{2} \left|\det\begin{pmatrix} 1 & 1 & 1 \\ P_x & Q_x & R_x \\ P_y & Q_y & R_y \end{pmatrix}\right|

Remark 6.1: Geometric Applications

Determinants are used in:

Computing areas and volumes in computational geometry
Testing for collinearity/coplanarity
Finding equations of lines/planes through given points
Change of variables in multiple integrals

Course 8 Practice Quiz

Questions

Correct

Accuracy

The determinant of

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

is:

Easy

Not attempted

\det(AB) =

Easy

Not attempted

\det(A^T) =

Easy

Not attempted

A

is invertible,

\det(A^{-1}) =

Medium

Not attempted

Determinant of an upper triangular matrix equals:

Easy

Not attempted

Laplace expansion along row

i

gives:

Medium

Not attempted

The adjugate matrix adj(A) satisfies:

Medium

Not attempted

Cramer's rule solves

Ax = b

when:

Easy

Not attempted

\det(cA)

for an

n \times n

matrix

A

equals:

Medium

Not attempted

If a matrix has two identical rows, then:

Easy

Not attempted

Frequently Asked Questions

What is the geometric meaning of the determinant?

For a 2×2 matrix, det(A) is the signed area of the parallelogram spanned by the columns. For 3×3, it's the signed volume of the parallelepiped. The sign indicates orientation (right-handed vs left-handed).

Why does det(AB) = det(A)det(B)?

This follows from the axiomatic definition. The map A ↦ det(A) is a multiplicative function. Geometrically, composing linear maps multiplies the volume scaling factors.

When should I use row reduction vs cofactor expansion?

Row reduction is O(n³) and preferred for large matrices. Cofactor expansion is O(n!) but useful for symbolic computation, small matrices, or when a row/column has many zeros.

What is the adjugate matrix and why is it important?

The adjugate adj(A) is the transpose of the cofactor matrix. It satisfies A·adj(A) = det(A)·I, giving the inverse formula A^{-1} = adj(A)/det(A) when det(A) ≠ 0.

How does Cramer's rule work?

For Ax = b with det(A) ≠ 0, Cramer's rule gives x_i = det(A_i)/det(A) where A_i is A with column i replaced by b. It's elegant but computationally expensive (O(n·n!)).