Determinants and Matrix Properties

Summary

Determinants reveal fundamental properties of matrices and transformations, with applications in solving systems and computing volumes. This chapter covers determinant computation methods, their properties, and geometric interpretation as the volume scaling factor of a transformation. You will also learn Cramer's rule and understand the relationship between determinants and matrix invertibility.

Concepts Covered

This chapter covers the following 13 concepts from the learning graph:

Determinant
2x2 Determinant
3x3 Determinant
Cofactor Expansion
Minor
Cofactor
Determinant Properties
Multiplicative Property
Transpose Determinant
Singular Matrix
Volume Scaling Factor
Signed Area
Cramers Rule

Prerequisites

This chapter builds on concepts from:

Introduction

The determinant is one of the most important numbers associated with a square matrix. While its definition might seem algebraically arbitrary at first, the determinant has profound geometric meaning: it measures how a linear transformation scales area (in 2D) or volume (in 3D and higher dimensions). A determinant of 2 means the transformation doubles all areas; a determinant of 0 means the transformation collapses space onto a lower dimension.

Understanding determinants connects algebraic matrix operations to geometric intuition about transformations. When you compute a determinant, you're answering the question: "How much does this transformation stretch or compress space?" This perspective makes determinants essential for computer graphics, physics simulations, and understanding when systems of equations have unique solutions.

Geometric Motivation: Signed Area and Volume

Before diving into formulas, let's build intuition for what determinants measure geometrically.

Signed Area in 2D

Consider two vectors \(\mathbf{u} = \begin{bmatrix} a \\ c \end{bmatrix}\) and \(\mathbf{v} = \begin{bmatrix} b \\ d \end{bmatrix}\) in 2D. These vectors form a parallelogram. The signed area of this parallelogram is:

\[\text{Signed Area} = ad - bc\]

The "signed" aspect captures orientation:

Positive area: \(\mathbf{v}\) is counterclockwise from \(\mathbf{u}\)
Negative area: \(\mathbf{v}\) is clockwise from \(\mathbf{u}\)
Zero area: vectors are parallel (parallelogram collapses to a line)

Configuration	Signed Area	Interpretation
CCW orientation	Positive	Preserves orientation
CW orientation	Negative	Reverses orientation
Parallel vectors	Zero	Collapses to lower dimension

Diagram: Signed Area Visualizer

Run the Signed Area Visualizer Fullscreen

Signed Area Interactive Visualizer

Type: microsim

Bloom Level: Understand (L2) Bloom Verb: interpret, explain

Learning Objective: Help students visualize the signed area of the parallelogram formed by two vectors, understanding how orientation affects the sign.

Canvas layout: - Main area: 2D coordinate plane with vectors and parallelogram - Right panel: Area calculation display - Bottom: Controls

Visual elements: - Vector u (red arrow) from origin - Vector v (blue arrow) from origin - Parallelogram shaded (green if positive, red if negative area) - Grid background for reference - Area value displayed prominently - Sign indicator (+/-) with color coding

Interactive controls: - Draggable endpoints for vectors u and v - Checkbox: Show parallelogram - Checkbox: Show area calculation formula - Button: Reset to default vectors - Display: Current signed area value

Default parameters: - u = (2, 1) - v = (1, 2) - Show parallelogram: true

Behavior: - Parallelogram updates in real-time as vectors are dragged - Area calculation shows ad - bc with current values - Color changes from green to red when area becomes negative - When area approaches zero, parallelogram visibly flattens

Implementation: p5.js with interactive vector dragging

Volume Scaling Factor

When a matrix \(\mathbf{A}\) represents a linear transformation, the determinant \(\det(\mathbf{A})\) tells us how the transformation scales volumes:

\(|\det(\mathbf{A})| > 1\): Transformation expands volumes
\(|\det(\mathbf{A})| < 1\): Transformation compresses volumes
\(|\det(\mathbf{A})| = 1\): Transformation preserves volumes
\(\det(\mathbf{A}) < 0\): Transformation also flips orientation
\(\det(\mathbf{A}) = 0\): Transformation collapses dimension

The absolute value \(|\det(\mathbf{A})|\) gives the volume scaling factor, while the sign indicates whether orientation is preserved or reversed.

Volume Scaling Examples

Rotation matrices have \(\det(\mathbf{R}) = 1\) (preserve area, preserve orientation)
Reflection matrices have \(\det(\mathbf{F}) = -1\) (preserve area, reverse orientation)
Scaling by factor \(k\) in all directions: \(\det = k^n\) where \(n\) is dimension
Projection matrices have \(\det = 0\) (collapse to lower dimension)

The 2×2 Determinant

The determinant of a 2×2 matrix is defined as:

\[\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc\]

This formula directly gives the signed area of the parallelogram formed by the column vectors. We also write \(\det(\mathbf{A})\) as \(|\mathbf{A}|\).

Computing 2×2 Determinants

The pattern is simple: multiply along the main diagonal, subtract the product along the anti-diagonal.

\[\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \underbrace{a \cdot d}_{\text{main diagonal}} - \underbrace{b \cdot c}_{\text{anti-diagonal}}\]

Examples:

\[\det\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix} = (3)(4) - (2)(1) = 12 - 2 = 10\]

\[\det\begin{bmatrix} 2 & 6 \\ 1 & 3 \end{bmatrix} = (2)(3) - (6)(1) = 6 - 6 = 0\]

The second example has determinant zero—the columns \((2, 1)\) and \((6, 3)\) are parallel (one is a scalar multiple of the other).

Diagram: 2×2 Determinant Calculator

Run the 2×2 Determinant Calculator Fullscreen

2×2 Determinant Interactive Calculator

Type: microsim

Bloom Level: Apply (L3) Bloom Verb: calculate, demonstrate

Learning Objective: Enable students to practice computing 2×2 determinants and see the geometric interpretation simultaneously.

Canvas layout: - Left: Matrix input area with editable cells - Center: Geometric visualization (parallelogram) - Right: Step-by-step calculation - Bottom: Controls

Visual elements: - 2×2 matrix with editable numerical entries - Parallelogram formed by column vectors - Main diagonal highlighted (green) - Anti-diagonal highlighted (red) - Area shading with sign indication - Step-by-step calculation: ad = ?, bc = ?, ad - bc = ?

Interactive controls: - Editable matrix entries (click to edit, -10 to 10 range) - Button: Random matrix - Button: Identity matrix - Button: Singular matrix (det = 0) - Checkbox: Show calculation steps - Checkbox: Animate diagonal products

Default parameters: - Matrix: [[3, 1], [2, 4]] - Show calculation: true

Behavior: - Parallelogram updates as matrix entries change - Calculation steps update in real-time - Color changes for positive/negative/zero determinant - Animation highlights diagonal products when enabled

Implementation: p5.js with text input handling

The 3×3 Determinant

For a 3×3 matrix, the determinant represents the signed volume of the parallelepiped (3D parallelogram) formed by the three column vectors.

\[\det\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)\]

The Rule of Sarrus

A mnemonic for 3×3 determinants is the Rule of Sarrus. Copy the first two columns to the right, then:

Add products along the three downward diagonals
Subtract products along the three upward diagonals

\[\det(\mathbf{A}) = aei + bfg + cdh - ceg - afh - bdi\]

While Sarrus' rule only works for 3×3 matrices (not larger ones), it provides a quick calculation method.

Diagram: 3×3 Determinant Sarrus Visualizer

Run the Rule of Sarrus Visualizer Fullscreen

Rule of Sarrus Interactive Visualizer

Type: microsim

Bloom Level: Apply (L3) Bloom Verb: execute, calculate

Learning Objective: Help students master the Rule of Sarrus for computing 3×3 determinants through visual step-by-step animation.

Canvas layout: - Main area: 3×3 matrix with extended columns - Diagonal lines shown during calculation - Running total display - Bottom: Controls

Visual elements: - 3×3 matrix grid with values - First two columns repeated to the right - Downward diagonal lines (green, positive terms) - Upward diagonal lines (red, negative terms) - Product values along each diagonal - Running total that builds up the determinant

Interactive controls: - Editable matrix entries - Button: Step through calculation - Button: Play animation - Button: Reset - Slider: Animation speed - Dropdown: Example matrices (identity, rotation, random)

Default parameters: - Matrix: [[1, 2, 3], [4, 5, 6], [7, 8, 9]] - Animation speed: medium

Behavior: - Step mode highlights one diagonal at a time - Shows product calculation for each diagonal - Running total updates with each step - Final result highlighted when complete - Color coding for positive (green) and negative (red) terms

Implementation: p5.js with step-based animation

Cofactor Expansion

For matrices larger than 3×3, we use cofactor expansion (also called Laplace expansion). This recursive method expresses a determinant in terms of smaller determinants.

Minors and Cofactors

For a matrix \(\mathbf{A}\), the minor \(M_{ij}\) is the determinant of the submatrix obtained by deleting row \(i\) and column \(j\).

The cofactor \(C_{ij}\) includes a sign factor:

\[C_{ij} = (-1)^{i+j} M_{ij}\]

The sign pattern alternates in a checkerboard fashion:

\[\begin{bmatrix} + & - & + & - & \cdots \\ - & + & - & + & \cdots \\ + & - & + & - & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix}\]

Cofactor Expansion Formula

The determinant can be computed by expanding along any row or column:

Expansion along row \(i\): \(\(\det(\mathbf{A}) = \sum_{j=1}^{n} a_{ij} C_{ij} = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} M_{ij}\)\)

Expansion along column \(j\): \(\(\det(\mathbf{A}) = \sum_{i=1}^{n} a_{ij} C_{ij} = \sum_{i=1}^{n} (-1)^{i+j} a_{ij} M_{ij}\)\)

The result is the same regardless of which row or column you choose. For efficiency, expand along a row or column with the most zeros.

Example: Compute \(\det\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}\) by expanding along column 1:

\[\det = 1 \cdot C_{11} + 0 \cdot C_{21} + 0 \cdot C_{31} = 1 \cdot (+1) \cdot \det\begin{bmatrix} 4 & 5 \\ 0 & 6 \end{bmatrix} = 1 \cdot (24 - 0) = 24\]

Efficiency Tip

Always expand along the row or column with the most zeros. This minimizes the number of cofactor calculations needed.

Diagram: Cofactor Expansion Step-by-Step

Run the Cofactor Expansion Visualizer Fullscreen

Cofactor Expansion Interactive Visualizer

Type: microsim

Bloom Level: Apply (L3) Bloom Verb: execute, implement

Learning Objective: Guide students through the cofactor expansion process step-by-step, showing how minors and cofactors combine to compute the determinant.

Canvas layout: - Top: Original matrix with selectable expansion row/column - Middle: Submatrices (minors) being calculated - Bottom: Cofactor combination and final result - Controls panel

Visual elements: - Matrix with highlighted expansion row/column - Sign checkerboard pattern overlay - Submatrices shown when computing each minor - Term-by-term calculation: a_ij × C_ij = ? - Summation showing all terms combining - Color coding: positive terms (green), negative terms (red)

Interactive controls: - Matrix size selector (3×3, 4×4) - Editable matrix entries - Dropdown: Select row or column for expansion - Button: Step through expansion - Button: Auto-play animation - Button: Reset - Checkbox: Show sign pattern

Default parameters: - Matrix size: 3×3 - Matrix: [[2, 1, 3], [4, 5, 6], [7, 8, 9]] - Expand along: row 1

Behavior: - Highlights selected row/column for expansion - Shows each minor being computed - Displays sign factor for each cofactor - Accumulates partial sums - Final determinant displayed with verification

Implementation: p5.js with step-based animation and submatrix visualization

Properties of Determinants

Determinants satisfy several important properties that simplify calculations and reveal structural information about matrices.

Key Determinant Properties

Property	Statement	Implication
Row swap	Swapping two rows negates the determinant	\(\det(\text{swap}) = -\det(\mathbf{A})\)
Row scaling	Multiplying a row by \(k\) multiplies det by \(k\)	\(\det(k\mathbf{A}) = k^n \det(\mathbf{A})\)
Row addition	Adding a multiple of one row to another doesn't change det	Useful for Gaussian elimination
Triangular	Determinant of triangular matrix = product of diagonal	Fast computation
Zero row/column	If any row or column is all zeros, det = 0	Matrix is singular
Proportional rows	If two rows are proportional, det = 0	Linear dependence

Multiplicative Property

One of the most important properties: the determinant of a product equals the product of determinants.

\[\det(\mathbf{A}\mathbf{B}) = \det(\mathbf{A}) \cdot \det(\mathbf{B})\]

Geometric interpretation: If transformation \(\mathbf{A}\) scales volume by factor \(|\det(\mathbf{A})|\) and transformation \(\mathbf{B}\) scales by \(|\det(\mathbf{B})|\), then the composition scales by \(|\det(\mathbf{A})| \cdot |\det(\mathbf{B})|\).

Consequences:

\(\det(\mathbf{A}^{-1}) = \frac{1}{\det(\mathbf{A})}\) (if \(\mathbf{A}\) is invertible)
\(\det(\mathbf{A}^n) = (\det(\mathbf{A}))^n\)
\(\det(\mathbf{I}) = 1\)

Transpose Determinant

The determinant is unchanged by transposition:

\[\det(\mathbf{A}^T) = \det(\mathbf{A})\]

This means every property that holds for rows also holds for columns. If two columns are proportional, the determinant is zero. If you scale a column by \(k\), the determinant scales by \(k\).

Diagram: Determinant Properties Explorer

Run the Determinant Properties Explorer Fullscreen

Determinant Properties Interactive Explorer

Type: microsim

Bloom Level: Analyze (L4) Bloom Verb: examine, differentiate

Learning Objective: Allow students to experiment with row/column operations and observe how each operation affects the determinant, building intuition for determinant properties.

Canvas layout: - Left: Original matrix with determinant - Center: Operation selector and visualization - Right: Modified matrix with new determinant - Bottom: Property explanation panel

Visual elements: - Two matrices side by side (before/after operation) - Determinant values displayed for both - Parallelogram visualizations (for 2×2 mode) - Ratio of determinants shown - Operation description text

Interactive controls: - Matrix size selector (2×2, 3×3) - Editable matrix entries - Operation buttons: - Swap rows i and j - Scale row i by k - Add k × row i to row j - Transpose - Multiply by another matrix - Slider: Scale factor k (-5 to 5) - Row/column selectors

Default parameters: - Matrix size: 2×2 - Matrix: [[2, 1], [3, 4]] - Operation: none selected

Behavior: - Shows before/after matrices - Displays determinant change: det(A) → det(A') - Explains the property being demonstrated - For 2×2, shows parallelogram area change - Highlights which property is being used

Implementation: p5.js with operation buttons and real-time calculation

Singular Matrices

A square matrix \(\mathbf{A}\) is singular (non-invertible) if and only if \(\det(\mathbf{A}) = 0\).

Characterizations of Singular Matrices

The following are equivalent for an \(n \times n\) matrix \(\mathbf{A}\):

\(\det(\mathbf{A}) = 0\)
\(\mathbf{A}\) is not invertible
\(\mathbf{A}\mathbf{x} = \mathbf{0}\) has non-trivial solutions
The columns of \(\mathbf{A}\) are linearly dependent
The rows of \(\mathbf{A}\) are linearly dependent
\(\text{rank}(\mathbf{A}) < n\)
The transformation collapses space to lower dimension
Zero is an eigenvalue of \(\mathbf{A}\)

Singular Matrices in Applications

In numerical computing, matrices with determinants close to zero (nearly singular) cause problems. Small errors get amplified, leading to unreliable results. The condition number measures how close a matrix is to being singular.

Geometric Interpretation of Singularity

When \(\det(\mathbf{A}) = 0\), the linear transformation represented by \(\mathbf{A}\) collapses at least one dimension:

In 2D: A plane gets squashed onto a line or point
In 3D: A volume gets flattened onto a plane, line, or point

This explains why singular matrices aren't invertible—once you collapse a dimension, you can't recover the original information.

Diagram: Singular vs Non-Singular Transformation

Run the Singular Matrix Visualizer Fullscreen

Singular vs Non-Singular Matrix Visualizer

Type: microsim

Bloom Level: Understand (L2) Bloom Verb: explain, interpret

Learning Objective: Visualize the geometric difference between singular (det=0) and non-singular matrices, showing how singular matrices collapse dimension.

Canvas layout: - Left: Original unit square with grid - Right: Transformed shape - Bottom: Matrix display and determinant - Controls panel

Visual elements: - Unit square (original) with grid points - Transformed parallelogram or collapsed line - Column vectors highlighted - Determinant value prominently displayed - "Singular" or "Non-singular" label - Animation showing transformation process

Interactive controls: - Editable 2×2 matrix entries - Slider: Interpolate between identity and current matrix - Button: Set to singular example (e.g., [[2,4],[1,2]]) - Button: Set to non-singular example - Button: Random matrix - Checkbox: Show grid transformation

Default parameters: - Matrix: [[2, 1], [4, 2]] (singular) - Animation: at full transformation

Behavior: - When det ≈ 0, shape collapses to line - Color changes to indicate singularity - Animation shows smooth collapse process - Explains why inverse doesn't exist geometrically - Shows rank deficiency visually

Implementation: p5.js with smooth animation

Volume Scaling in Higher Dimensions

The determinant generalizes the concept of signed area to higher dimensions:

Dimension	Geometric Object	Determinant Measures
2D	Parallelogram	Signed area
3D	Parallelepiped	Signed volume
nD	n-dimensional parallelotope	Signed hypervolume

Computing Volumes with Determinants

Given vectors \(\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\) in \(\mathbb{R}^n\), the (signed) volume of the parallelepiped they span is:

\[\text{Volume} = \det\begin{bmatrix} | & | & & | \\ \mathbf{v}_1 & \mathbf{v}_2 & \cdots & \mathbf{v}_n \\ | & | & & | \end{bmatrix}\]

For the unsigned volume, take the absolute value: \(|\det(\mathbf{A})|\).

Application: In computer graphics, determinants help determine:

Whether a set of points is coplanar (det = 0)
The orientation of a triangle (clockwise vs counterclockwise)
The volume of a tetrahedron for collision detection

Diagram: 3D Volume Scaling Visualizer

Run the 3D Volume Scaling Visualizer Fullscreen

3D Volume Scaling Interactive Visualizer

Type: microsim

Bloom Level: Understand (L2) Bloom Verb: explain, interpret

Learning Objective: Help students visualize how 3×3 matrix transformations scale 3D volumes, connecting the determinant to geometric volume change.

Canvas layout: - Main area: 3D view with unit cube and transformed parallelepiped - Right panel: Matrix and determinant display - Bottom: Controls

Visual elements: - Unit cube (wireframe, semi-transparent) - Transformed parallelepiped (solid, colored by det sign) - Three column vectors as arrows from origin - Axes with labels - Volume ratio display: V'/V = |det(A)| - Determinant value with sign

Interactive controls: - 3×3 matrix editor - Camera rotation (click and drag) - Slider: Animation morph (0 = identity, 1 = full transform) - Button: Rotation matrix example - Button: Scaling matrix example - Button: Singular matrix example - Checkbox: Show original cube - Checkbox: Show column vectors

Default parameters: - Matrix: [[2,0,0],[0,1.5,0],[0,0,1]] (scaling) - Animation: at 1.0 - Camera angle: isometric view

Behavior: - Real-time 3D rendering with rotation - Smooth morphing animation - Volume label updates with determinant - For singular matrices, parallelepiped collapses - Color indicates positive (green) or negative (red) determinant

Implementation: p5.js with WEBGL for 3D rendering

Cramer's Rule

Cramer's Rule provides explicit formulas for solving systems of linear equations using determinants.

The Formula

For a system \(\mathbf{A}\mathbf{x} = \mathbf{b}\) where \(\mathbf{A}\) is \(n \times n\) and \(\det(\mathbf{A}) \neq 0\):

\[x_i = \frac{\det(\mathbf{A}_i)}{\det(\mathbf{A})}\]

where \(\mathbf{A}_i\) is the matrix formed by replacing column \(i\) of \(\mathbf{A}\) with \(\mathbf{b}\).

Example: 2×2 System

Solve: \(\begin{cases} 2x + 3y = 7 \\ x + 4y = 9 \end{cases}\)

Matrix form: \(\mathbf{A} = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}\), \(\mathbf{b} = \begin{bmatrix} 7 \\ 9 \end{bmatrix}\)

Step 1: \(\det(\mathbf{A}) = 2(4) - 3(1) = 5\)

Step 2: Replace column 1 with \(\mathbf{b}\): \(\(\det(\mathbf{A}_1) = \det\begin{bmatrix} 7 & 3 \\ 9 & 4 \end{bmatrix} = 28 - 27 = 1\)\)

Step 3: Replace column 2 with \(\mathbf{b}\): \(\(\det(\mathbf{A}_2) = \det\begin{bmatrix} 2 & 7 \\ 1 & 9 \end{bmatrix} = 18 - 7 = 11\)\)

Solution: \(x = \frac{1}{5}\), \(y = \frac{11}{5}\)

When to Use Cramer's Rule

Situation	Recommendation
Small systems (2×2, 3×3)	Cramer's rule is practical
Solve for one variable only	Efficient—only compute relevant det
Large systems	Use Gaussian elimination (more efficient)
Theoretical analysis	Cramer's rule gives explicit formulas
\(\det(\mathbf{A}) = 0\)	Cramer's rule doesn't apply

Computational Efficiency

For large systems, Cramer's rule requires computing \(n+1\) determinants, each of which is \(O(n!)\) by cofactor expansion or \(O(n^3)\) by LU decomposition. Gaussian elimination solves the system in \(O(n^3)\), making it much more efficient for large \(n\).

Diagram: Cramer's Rule Step-by-Step Solver

Run the Cramer's Rule Solver Fullscreen

Cramer's Rule Interactive Solver

Type: microsim

Bloom Level: Apply (L3) Bloom Verb: solve, calculate

Learning Objective: Guide students through applying Cramer's rule to solve 2×2 and 3×3 systems, showing each replacement and determinant calculation.

Canvas layout: - Top: Original system of equations - Middle: Step-by-step determinant calculations - Bottom: Solution display and geometric interpretation (for 2×2) - Controls panel

Visual elements: - System of equations in standard form - Matrix A and vector b displayed - Each modified matrix A_i highlighted - Determinant calculations shown - Solution vector - For 2×2: geometric interpretation showing intersection of lines

Interactive controls: - System size selector (2×2, 3×3) - Editable coefficient matrix A - Editable vector b - Button: Step through solution - Button: Auto-solve with animation - Button: Random system - Button: Singular system (to show failure case)

Default parameters: - System size: 2×2 - A = [[2, 3], [1, 4]] - b = [7, 9]

Behavior: - Highlights which column is being replaced - Shows each determinant calculation - Builds up solution step by step - If det(A) = 0, shows "No unique solution" message - Geometric view shows lines and their intersection

Implementation: p5.js with step-based animation

Computing Determinants Efficiently

For practical computation, several methods are available:

Method Comparison

Method	Time Complexity	Best For
Direct formula	\(O(1)\) for 2×2	2×2 matrices
Sarrus rule	\(O(1)\)	3×3 matrices
Cofactor expansion	\(O(n!)\)	Small matrices, matrices with many zeros
Row reduction	\(O(n^3)\)	Large matrices
LU decomposition	\(O(n^3)\)	Multiple determinants, also need inverse

Row Reduction Method

Reduce \(\mathbf{A}\) to upper triangular form \(\mathbf{U}\) using row operations
Track row swaps (each negates the determinant)
Track row scalings (each multiplies the determinant)
\(\det(\mathbf{A}) = (\text{sign from swaps}) \times (\text{scaling factors}) \times \prod_{i} u_{ii}\)

For an upper triangular matrix, the determinant is simply the product of diagonal entries:

\[\det(\mathbf{U}) = u_{11} \cdot u_{22} \cdot \ldots \cdot u_{nn}\]

Applications

Computer Graphics

Determinants are used to:

Test if three points are collinear (2×2 det = 0)
Determine triangle orientation for rendering
Calculate areas for texture mapping
Check if a transformation preserves handedness

Physics and Engineering

Calculate moments of inertia (involving volume integrals)
Solve systems in circuit analysis
Determine stability of equilibrium points
Compute Jacobians for coordinate transformations

Machine Learning

Check if features are linearly independent
Compute multivariate Gaussian distributions
Calculate volume elements in probability spaces
Understand transformation properties in neural networks

Summary and Key Takeaways

This chapter explored determinants as fundamental matrix quantities with both algebraic and geometric significance.

Core Concepts:

The determinant measures how a transformation scales volumes
Signed area (2D) and volume scaling factor (3D+) connect determinants to geometry
The 2×2 determinant is \(ad - bc\) (main diagonal minus anti-diagonal)
The 3×3 determinant can be computed via Sarrus' rule or cofactor expansion
Cofactor expansion generalizes to any size using minors and cofactors

Key Properties:

Multiplicative property: \(\det(\mathbf{AB}) = \det(\mathbf{A})\det(\mathbf{B})\)
Transpose property: \(\det(\mathbf{A}^T) = \det(\mathbf{A})\)
Row operations affect determinants predictably
Triangular matrices have det = product of diagonal

Singularity:

\(\det(\mathbf{A}) = 0\) means \(\mathbf{A}\) is singular (not invertible)
Geometrically: transformation collapses dimension
Algebraically: columns are linearly dependent

Applications:

Cramer's Rule solves systems using determinant ratios
Efficient computation uses row reduction for large matrices
Determinants appear throughout graphics, physics, and ML

Exercises

Exercise 1: Computing Determinants

Calculate the determinant of each matrix:

a) \(\begin{bmatrix} 5 & 2 \\ 3 & 4 \end{bmatrix}\)

b) \(\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}\)

c) \(\begin{bmatrix} 2 & 1 & 0 \\ 1 & 3 & 2 \\ 0 & 1 & 4 \end{bmatrix}\)

Exercise 2: Determinant Properties

If \(\det(\mathbf{A}) = 3\) and \(\det(\mathbf{B}) = -2\), find:

a) \(\det(\mathbf{AB})\)

b) \(\det(\mathbf{A}^{-1})\)

c) \(\det(2\mathbf{A})\) for a 3×3 matrix

d) \(\det(\mathbf{A}^T\mathbf{B})\)

Exercise 3: Cramer's Rule

Use Cramer's rule to solve: \(\begin{cases} 3x + 2y = 8 \\ x - y = 1 \end{cases}\)

Exercise 4: Geometric Interpretation

The transformation \(T\) has matrix \(\mathbf{A} = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}\).

a) What is \(\det(\mathbf{A})\)?

b) If a triangle has area 5, what is the area of its image under \(T\)?

c) Is the transformation orientation-preserving?

Exercise 5: Singularity

For what values of \(k\) is the matrix singular? \(\(\mathbf{A} = \begin{bmatrix} 1 & 2 & k \\ 0 & k-1 & 3 \\ 0 & 0 & k+2 \end{bmatrix}\)\)