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<!DOCTYPE html>
<html lang="en">
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<title>Matrix & Vectors — Complete Guide</title>
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<div class="header-title">Matrix & Vectors</div>
<div class="header-badge">Complete Reference</div>
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<div class="sidebar-section-title">Matrix Fundamentals</div>
<a class="sidebar-item" href="#mat-intro"><span class="dot"></span>What is a Matrix</a>
<a class="sidebar-item" href="#mat-types"><span class="dot"></span>Types of Matrices</a>
<a class="sidebar-item" href="#mat-ops"><span class="dot"></span>Matrix Operations</a>
<a class="sidebar-item" href="#mat-det"><span class="dot"></span>Determinant</a>
<a class="sidebar-item" href="#mat-inv"><span class="dot"></span>Matrix Inverse</a>
<a class="sidebar-item" href="#mat-rank"><span class="dot"></span>Rank of a Matrix</a>
<a class="sidebar-item" href="#mat-transpose"><span class="dot"></span>Transpose & Symmetric</a>
<a class="sidebar-item" href="#mat-system"><span class="dot"></span>System of Equations</a>
<div class="sidebar-section-title">Advanced Matrix</div>
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<a class="sidebar-item" href="#vec-proj"><span class="dot"></span>Projection of a Vector</a>
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<a class="sidebar-item" href="#vec-sol"><span class="dot"></span>Solenoidal <span class="star">★★★★★</span></a>
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<a class="sidebar-item" href="#vec-irr"><span class="dot"></span>Irrotational <span class="star">★★★★★</span></a>
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<div class="hero-eyebrow">Advanced Mathematics</div>
<h1 class="hero-title">Matrix &<br>Vectors</h1>
<p class="hero-sub">A complete, structured reference covering matrix theory, vector algebra, differential operations, and integral theorems — with clear concepts and worked examples.</p>
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<span class="tag tag-blue">Carley Hamington</span>
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<!-- ══════════════════════════════════════
CHAPTER 0: MATRIX FUNDAMENTALS
══════════════════════════════════════ -->
<div class="chapter-group">
<div class="chapter-banner">
<div class="chapter-icon">⬚</div>
<div>
<div class="chapter-name">Chapter 0 — Matrix Fundamentals</div>
<div class="chapter-desc">Definition, Types, Operations, Determinants, Inverse, Rank, and Linear Systems</div>
</div>
</div>
<!-- ── WHAT IS A MATRIX ── -->
<section class="topic-section" id="mat-intro">
<div class="section-header">
<div class="section-number">00</div>
<h2 class="section-title">What is a <span class="highlight">Matrix</span>?</h2>
</div>
<div class="def-box">
<span class="def-label">Definition</span>
A <strong>matrix</strong> is a rectangular array of numbers, symbols, or expressions arranged in <strong>rows</strong> and <strong>columns</strong>. A matrix with m rows and n columns is called an <strong>m × n matrix</strong>. The element at row i and column j is denoted <strong>aᵢⱼ</strong>. Matrices are used to represent linear transformations, systems of equations, and data structures in engineering and science.
</div>
<div class="concept-card">
<div class="concept-label">Concept</div>
<div class="concept-title">General Form of a Matrix</div>
<div class="concept-desc">
An m × n matrix A is written as:<br><br>
<span class="formula">A = [aᵢⱼ]ₘₓₙ</span><br><br>
where i = 1,2,...,m (rows) and j = 1,2,...,n (columns). A <strong>square matrix</strong> has m = n. The main diagonal consists of elements a₁₁, a₂₂, ..., aₙₙ.
</div>
</div>
<div class="formula-box">
<span class="formula-label">Standard Form</span>
<div class="math-block"> ⎡ a₁₁ a₁₂ a₁₃ ⎤
A = ⎢ a₂₁ a₂₂ a₂₃ ⎥ (3×3 matrix)
⎣ a₃₁ a₃₂ a₃₃ ⎦
Order = m × n (m rows, n columns)
Element at row i, col j = aᵢⱼ</div>
</div>
<ul class="prop-list">
<li>Two matrices are equal if and only if they have the same order and every corresponding element is equal.</li>
<li>A matrix is a tool — it encodes transformations, not just data.</li>
<li>Row matrix: 1×n | Column matrix: m×1 | Square: m=n</li>
<li>The size (order) of a matrix is always stated as rows × columns.</li>
</ul>
<div class="examples-grid">
<div class="example-card">
<div class="example-header"><span class="example-num">EX 1</span><span class="example-title">Identify Order & Elements</span></div>
<div class="example-body">
<p>Given matrix A, find its order and element a₂₃.</p>
<div class="math-block">A = ⎡ 1 2 3 ⎤
⎢ 4 5 6 ⎥
⎣ 7 8 9 ⎦</div>
<div class="result-line">Order = 3×3, a₂₃ = 6</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 2</span><span class="example-title">Row & Column Matrix</span></div>
<div class="example-body">
<p>Identify the types of these matrices.</p>
<div class="math-block">R = [3 7 −2] → Row (1×3)
⎡ 5 ⎤
C = ⎢ −1 ⎥ → Column (3×1)
⎣ 8 ⎦</div>
<div class="result-line">R is 1×3, C is 3×1</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 3</span><span class="example-title">Matrix Equality</span></div>
<div class="example-body">
<p>Find x, y if [2x, 3] = [6, y+1]</p>
<div class="math-block">Corresponding elements equal:
2x = 6 → x = 3
3 = y+1 → y = 2</div>
<div class="result-line">x = 3, y = 2</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 4</span><span class="example-title">Construct a Matrix</span></div>
<div class="example-body">
<p>Construct 2×3 matrix where aᵢⱼ = i + 2j</p>
<div class="math-block">a₁₁=3 a₁₂=5 a₁₃=7
a₂₁=4 a₂₂=6 a₂₃=8
A = ⎡ 3 5 7 ⎤
⎣ 4 6 8 ⎦</div>
<div class="result-line">2×3 matrix constructed</div>
</div>
</div>
</div>
</section>
<!-- ── TYPES OF MATRICES ── -->
<section class="topic-section" id="mat-types">
<div class="section-header">
<div class="section-number">00b</div>
<h2 class="section-title">Types of <span class="highlight">Matrices</span></h2>
</div>
<div class="def-box">
<span class="def-label">Definition</span>
Matrices are classified by their <strong>shape</strong>, <strong>properties</strong>, and <strong>element values</strong>. Knowing the type of a matrix helps predict its behavior in operations and theorems.
</div>
<div class="formula-grid">
<div class="formula-item">
<div class="fi-name">Zero / Null Matrix</div>
<div class="fi-eq">All elements = 0
O = [[0,0],[0,0]]</div>
</div>
<div class="formula-item">
<div class="fi-name">Identity Matrix</div>
<div class="fi-eq">Diagonal = 1, rest = 0
I = [[1,0],[0,1]]</div>
</div>
<div class="formula-item">
<div class="fi-name">Diagonal Matrix</div>
<div class="fi-eq">Non-zero only on diagonal
D = [[d₁,0],[0,d₂]]</div>
</div>
<div class="formula-item">
<div class="fi-name">Scalar Matrix</div>
<div class="fi-eq">Diagonal matrix with
equal diagonal elements</div>
</div>
<div class="formula-item">
<div class="fi-name">Upper Triangular</div>
<div class="fi-eq">aᵢⱼ = 0 for i > j
(zeros below diagonal)</div>
</div>
<div class="formula-item">
<div class="fi-name">Lower Triangular</div>
<div class="fi-eq">aᵢⱼ = 0 for i < j
(zeros above diagonal)</div>
</div>
<div class="formula-item">
<div class="fi-name">Symmetric Matrix</div>
<div class="fi-eq">A = Aᵀ (aᵢⱼ = aⱼᵢ)</div>
</div>
<div class="formula-item">
<div class="fi-name">Skew-Symmetric</div>
<div class="fi-eq">A = −Aᵀ (aᵢⱼ = −aⱼᵢ)
Diagonal must be zero</div>
</div>
<div class="formula-item">
<div class="fi-name">Orthogonal Matrix</div>
<div class="fi-eq">A · Aᵀ = I
(Aᵀ = A⁻¹)</div>
</div>
<div class="formula-item">
<div class="fi-name">Singular Matrix</div>
<div class="fi-eq">det(A) = 0
No inverse exists</div>
</div>
<div class="formula-item">
<div class="fi-name">Idempotent Matrix</div>
<div class="fi-eq">A² = A</div>
</div>
<div class="formula-item">
<div class="fi-name">Involutory Matrix</div>
<div class="fi-eq">A² = I (A = A⁻¹)</div>
</div>
</div>
<div class="examples-grid">
<div class="example-card">
<div class="example-header"><span class="example-num">EX 1</span><span class="example-title">Identify Matrix Type</span></div>
<div class="example-body">
<p>Classify A = [[4,0,0],[0,7,0],[0,0,2]]</p>
<div class="math-block">Non-zero only on diagonal
→ Diagonal Matrix
All diagonal values different
→ NOT scalar (scalars need d₁=d₂=d₃)</div>
<div class="result-line">Diagonal matrix (also square)</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 2</span><span class="example-title">Verify Symmetric</span></div>
<div class="example-body">
<p>A = [[1,2,3],[2,5,4],[3,4,6]] — Is A symmetric?</p>
<div class="math-block">Check aᵢⱼ = aⱼᵢ:
a₁₂=2=a₂₁ ✓
a₁₃=3=a₃₁ ✓
a₂₃=4=a₃₂ ✓</div>
<div class="result-line">Yes, A is symmetric</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 3</span><span class="example-title">Skew-Symmetric Check</span></div>
<div class="example-body">
<p>A = [[0,3,−2],[−3,0,5],[2,−5,0]]</p>
<div class="math-block">Diagonal = all 0 ✓
a₁₂ = 3, a₂₁ = −3 = −a₁₂ ✓
a₁₃ = −2, a₃₁ = 2 = −a₁₃ ✓</div>
<div class="result-line">Skew-symmetric matrix ✓</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 4</span><span class="example-title">Idempotent Verify</span></div>
<div class="example-body">
<p>A = [[1,0],[0,0]]. Verify A² = A.</p>
<div class="math-block">A² = [[1,0],[0,0]]·[[1,0],[0,0]]
= [[1,0],[0,0]] = A ✓</div>
<div class="result-line">A is idempotent</div>
</div>
</div>
</div>
</section>
<!-- ── MATRIX OPERATIONS ── -->
<section class="topic-section" id="mat-ops">
<div class="section-header">
<div class="section-number">00c</div>
<h2 class="section-title">Matrix <span class="highlight">Operations</span></h2>
</div>
<div class="def-box">
<span class="def-label">Definition</span>
<strong>Addition/Subtraction:</strong> Matrices of the <strong>same order</strong> are added/subtracted element-wise. <strong>Scalar Multiplication:</strong> Each element is multiplied by the scalar. <strong>Matrix Multiplication:</strong> (A·B)ᵢⱼ = Σ aᵢₖ bₖⱼ. For A(m×n) and B(n×p), the result C is m×p. <strong>Note:</strong> Matrix multiplication is NOT commutative — AB ≠ BA in general.
</div>
<div class="formula-box">
<span class="formula-label">Key Formulas</span>
<div class="math-block">Addition: (A+B)ᵢⱼ = aᵢⱼ + bᵢⱼ
Subtraction: (A−B)ᵢⱼ = aᵢⱼ − bᵢⱼ
Scalar mult: (kA)ᵢⱼ = k·aᵢⱼ
Matrix mult: (AB)ᵢⱼ = Σₖ aᵢₖ·bₖⱼ
A is m×n, B is n×p ⟹ AB is m×p</div>
</div>
<ul class="prop-list">
<li>A + B = B + A (commutative for addition)</li>
<li>A(BC) = (AB)C (associative for multiplication)</li>
<li>A(B+C) = AB + AC (distributive)</li>
<li>AB ≠ BA in general (not commutative)</li>
<li>AI = IA = A (identity is multiplicative identity)</li>
</ul>
<div class="examples-grid">
<div class="example-card">
<div class="example-header"><span class="example-num">EX 1</span><span class="example-title">Matrix Addition</span></div>
<div class="example-body">
<p>A = [[1,2],[3,4]], B = [[5,6],[7,8]]</p>
<div class="math-block">A + B = [[1+5, 2+6],
[3+7, 4+8]]
= [[6, 8],
[10, 12]]</div>
<div class="result-line">A+B = [[6,8],[10,12]]</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 2</span><span class="example-title">Scalar Multiplication</span></div>
<div class="example-body">
<p>3A where A = [[2,−1],[0,4]]</p>
<div class="math-block">3A = [[3×2, 3×(−1)],
[3×0, 3×4 ]]
= [[6, −3],
[0, 12]]</div>
<div class="result-line">3A = [[6,−3],[0,12]]</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 3</span><span class="example-title">Matrix Multiplication</span></div>
<div class="example-body">
<p>A = [[1,2],[3,4]], B = [[5,6],[7,8]]</p>
<div class="math-block">AB:
c₁₁ = 1×5+2×7 = 19
c₁₂ = 1×6+2×8 = 22
c₂₁ = 3×5+4×7 = 43
c₂₂ = 3×6+4×8 = 50</div>
<div class="result-line">AB = [[19,22],[43,50]]</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 4</span><span class="example-title">Non-Commutativity</span></div>
<div class="example-body">
<p>A=[[1,0],[2,1]], B=[[1,2],[0,1]]. Show AB ≠ BA.</p>
<div class="math-block">AB = [[1,2],[2,5]]
BA = [[5,2],[2,1]]
AB ≠ BA ✓ (not commutative)</div>
<div class="result-line">Matrix mult. is NOT commutative</div>
</div>
</div>
</div>
</section>
<!-- ── DETERMINANT ── -->
<section class="topic-section" id="mat-det">
<div class="section-header">
<div class="section-number">00d</div>
<h2 class="section-title"><span class="highlight">Determinant</span> of a Matrix</h2>
</div>
<div class="def-box">
<span class="def-label">Definition</span>
The <strong>determinant</strong> is a scalar value computed from a <strong>square matrix</strong> that encodes geometric and algebraic information about the matrix. It tells us whether the matrix is invertible (det ≠ 0) or singular (det = 0), and its absolute value represents the scale factor of the linear transformation.
</div>
<div class="formula-box">
<span class="formula-label">Formulas</span>
<div class="math-block">2×2: det(A) = |a b| = ad − bc
|c d|
3×3 (cofactor expansion along row 1):
|a b c|
|d e f| = a(ei−fh) − b(di−fg) + c(dh−eg)
|g h i|
Properties:
det(AB) = det(A)·det(B)
det(Aᵀ) = det(A)
det(kA) = kⁿ·det(A) for n×n matrix
det(A⁻¹) = 1/det(A)</div>
</div>
<div class="examples-grid">
<div class="example-card">
<div class="example-header"><span class="example-num">EX 1</span><span class="example-title">2×2 Determinant</span></div>
<div class="example-body">
<p>Find det(A) for A = [[3,8],[4,6]]</p>
<div class="math-block">det(A) = (3)(6) − (8)(4)
= 18 − 32
= −14</div>
<div class="result-line">det(A) = −14</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 2</span><span class="example-title">3×3 Determinant</span></div>
<div class="example-body">
<p>A = [[1,2,3],[4,5,6],[7,8,9]]</p>
<div class="math-block">= 1(5·9−6·8)−2(4·9−6·7)+3(4·8−5·7)
= 1(45−48)−2(36−42)+3(32−35)
= 1(−3)−2(−6)+3(−3)
= −3+12−9 = 0</div>
<div class="result-line">det = 0 (singular matrix)</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 3</span><span class="example-title">Using Cofactor Expansion</span></div>
<div class="example-body">
<p>A = [[2,1,0],[1,3,2],[0,1,4]]</p>
<div class="math-block">Expand along row 1:
= 2(3·4−2·1)−1(1·4−2·0)+0
= 2(10)−1(4)
= 20 − 4 = 16</div>
<div class="result-line">det(A) = 16 (invertible)</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 4</span><span class="example-title">Property: det(AB)</span></div>
<div class="example-body">
<p>A=[[1,2],[3,4]], B=[[5,6],[7,8]]</p>
<div class="math-block">det(A) = 4−6 = −2
det(B) = 40−42 = −2
det(AB) = (−2)(−2) = 4
Verify: AB=[[19,22],[43,50]]
det(AB) = 950−946 = 4 ✓</div>
</div>
</div>
</div>
</section>
<!-- ── MATRIX INVERSE ── -->
<section class="topic-section" id="mat-inv">
<div class="section-header">
<div class="section-number">00e</div>
<h2 class="section-title">Matrix <span class="highlight">Inverse</span></h2>
</div>
<div class="def-box">
<span class="def-label">Definition</span>
The <strong>inverse</strong> of a square matrix A, denoted <strong>A⁻¹</strong>, is the matrix such that <strong>A · A⁻¹ = A⁻¹ · A = I</strong> (identity matrix). The inverse exists <strong>if and only if det(A) ≠ 0</strong>. A matrix without an inverse is called <strong>singular</strong>.
</div>
<div class="formula-box">
<span class="formula-label">Formulas</span>
<div class="math-block">For 2×2 matrix A = [[a,b],[c,d]]:
A⁻¹ = 1/det(A) · [[d,−b],[−c,a]]
For n×n matrix (general):
A⁻¹ = (1/det(A)) · adj(A)
where adj(A) = transpose of cofactor matrix
Cofactor Cᵢⱼ = (−1)^(i+j) · Mᵢⱼ
Mᵢⱼ = minor (det of submatrix without row i, col j)</div>
</div>
<div class="examples-grid">
<div class="example-card">
<div class="example-header"><span class="example-num">EX 1</span><span class="example-title">2×2 Inverse</span></div>
<div class="example-body">
<p>Find A⁻¹ for A = [[4,7],[2,6]]</p>
<div class="math-block">det(A) = 24−14 = 10
A⁻¹ = (1/10)[[6,−7],[−2,4]]
= [[0.6, −0.7],
[−0.2, 0.4]]</div>
<div class="result-line">A⁻¹ = (1/10)[[6,−7],[−2,4]]</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 2</span><span class="example-title">Verify AA⁻¹ = I</span></div>
<div class="example-body">
<p>A=[[4,7],[2,6]], A⁻¹=(1/10)[[6,−7],[−2,4]]</p>
<div class="math-block">AA⁻¹ = (1/10)[[4,7],[2,6]]·[[6,−7],[−2,4]]
= (1/10)[[24−14, −28+28],
[12−12, −14+24]]
= (1/10)[[10,0],[0,10]]
= [[1,0],[0,1]] = I ✓</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 3</span><span class="example-title">3×3 Inverse via Adjoint</span></div>
<div class="example-body">
<p>A = [[2,1,0],[1,3,2],[0,1,4]]. We know det(A)=16.</p>
<div class="math-block">Cofactors: C₁₁=10, C₁₂=−4, C₁₃=1
C₂₁=−4, C₂₂=8, C₂₃=−2
C₃₁=2, C₃₂=−4, C₃₃=5
adj(A) = Cᵀ
A⁻¹ = (1/16)·adj(A)</div>
<div class="result-line">A⁻¹ = (1/16)·adj(A)</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 4</span><span class="example-title">Inverse via Row Reduction</span></div>
<div class="example-body">
<p>Augment [A|I] and row-reduce to get [I|A⁻¹]</p>
<div class="math-block">A = [[1,2],[3,4]]
[A|I] = [[1,2|1,0],[3,4|0,1]]
R2→R2−3R1:
[[1,2|1,0],[0,−2|−3,1]]
R2÷(−2):[[1,2|1,0],[0,1|3/2,−1/2]]
R1→R1−2R2:[[1,0|−2,1],[0,1|3/2,−1/2]]</div>
<div class="result-line">A⁻¹ = [[−2,1],[3/2,−1/2]]</div>
</div>
</div>
</div>
</section>
<!-- ── RANK ── -->
<section class="topic-section" id="mat-rank">
<div class="section-header">
<div class="section-number">00f</div>
<h2 class="section-title">Rank of a <span class="highlight">Matrix</span></h2>
</div>
<div class="def-box">
<span class="def-label">Definition</span>
The <strong>rank</strong> of a matrix A is the <strong>maximum number of linearly independent rows</strong> (or columns). It equals the number of non-zero rows in the row echelon form of A. For an m×n matrix: <strong>rank(A) ≤ min(m,n)</strong>. If rank = n for a square n×n matrix, then A is non-singular (invertible).
</div>
<div class="formula-box">
<span class="formula-label">Key Rules</span>
<div class="math-block">rank(A) = number of non-zero rows in REF
rank(A) ≤ min(m,n)
rank(A) = rank(Aᵀ)
Nullity = n − rank(A) (Rank-Nullity Theorem)
For square n×n matrix:
rank = n ↔ det(A) ≠ 0 ↔ A is invertible
rank < n ↔ det(A) = 0 ↔ A is singular</div>
</div>
<div class="examples-grid">
<div class="example-card">
<div class="example-header"><span class="example-num">EX 1</span><span class="example-title">Rank by Row Reduction</span></div>
<div class="example-body">
<p>Find rank of A = [[1,2,3],[4,5,6],[7,8,9]]</p>
<div class="math-block">R2→R2−4R1: [0,−3,−6]
R3→R3−7R1: [0,−6,−12]
R3→R3−2R2: [0,0,0]
Non-zero rows = 2</div>
<div class="result-line">rank(A) = 2</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 2</span><span class="example-title">Full Rank Matrix</span></div>
<div class="example-body">
<p>A = [[1,0,0],[0,2,0],[0,0,3]]</p>
<div class="math-block">Already in row echelon form.
All 3 rows are non-zero and
linearly independent.</div>
<div class="result-line">rank(A) = 3 (full rank)</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 3</span><span class="example-title">Rank via Minors</span></div>
<div class="example-body">
<p>A = [[1,2],[2,4]]. Find rank.</p>
<div class="math-block">det(A) = 4−4 = 0 (rank < 2)
Row 1 ≠ zero row (rank ≥ 1)
Check 2×2 minor: det = 0
Check any 1×1 minor: 1 ≠ 0</div>
<div class="result-line">rank(A) = 1</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 4</span><span class="example-title">Rank-Nullity Theorem</span></div>
<div class="example-body">
<p>A is 3×5 with rank 2. Find nullity.</p>
<div class="math-block">Rank-Nullity Theorem:
rank + nullity = n (columns)
2 + nullity = 5
nullity = 3</div>
<div class="result-line">Nullity = 3 (solution space dim)</div>
</div>
</div>
</div>
</section>
<!-- ── TRANSPOSE & SYMMETRIC ── -->
<section class="topic-section" id="mat-transpose">
<div class="section-header">
<div class="section-number">00g</div>
<h2 class="section-title">Transpose & <span class="highlight">Symmetric</span> Matrices</h2>
</div>
<div class="def-box">
<span class="def-label">Definition</span>
The <strong>transpose</strong> of a matrix A, written <strong>Aᵀ</strong>, is obtained by interchanging its rows and columns: (Aᵀ)ᵢⱼ = Aⱼᵢ. If A is m×n, then Aᵀ is n×m. A matrix is <strong>symmetric</strong> if A = Aᵀ, and <strong>skew-symmetric</strong> if A = −Aᵀ. Every square matrix can be written as A = S + K where S is symmetric and K is skew-symmetric.
</div>
<div class="formula-box">
<span class="formula-label">Formulas & Properties</span>
<div class="math-block">Transpose: (Aᵀ)ᵢⱼ = Aⱼᵢ
(Aᵀ)ᵀ = A
(A+B)ᵀ = Aᵀ + Bᵀ
(AB)ᵀ = BᵀAᵀ (order reverses!)
(kA)ᵀ = k·Aᵀ
det(Aᵀ) = det(A)
Symmetric decomposition:
S = (A + Aᵀ)/2 (symmetric part)
K = (A − Aᵀ)/2 (skew-symmetric part)
A = S + K</div>
</div>
<div class="examples-grid">
<div class="example-card">
<div class="example-header"><span class="example-num">EX 1</span><span class="example-title">Find Transpose</span></div>
<div class="example-body">
<p>A = [[1,2,3],[4,5,6]]</p>
<div class="math-block"> ⎡1 4⎤
Aᵀ = ⎢2 5⎥ (2×3 → 3×2)
⎣3 6⎦</div>
<div class="result-line">Aᵀ is 3×2 matrix</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 2</span><span class="example-title">Verify (AB)ᵀ = BᵀAᵀ</span></div>
<div class="example-body">
<p>A=[[1,2],[3,4]], B=[[5,6],[7,8]]</p>
<div class="math-block">AB=[[19,22],[43,50]]
(AB)ᵀ=[[19,43],[22,50]]
BᵀAᵀ=[[5,7],[6,8]]·[[1,3],[2,4]]
=[[19,43],[22,50]] ✓</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 3</span><span class="example-title">Symmetric Decomposition</span></div>
<div class="example-body">
<p>A = [[3,1],[5,2]]. Decompose into S+K.</p>
<div class="math-block">S = (A+Aᵀ)/2 = [[3,3],[3,2]]
K = (A−Aᵀ)/2 = [[0,−2],[2,0]]
S+K = [[3,1],[5,2]] = A ✓</div>
<div class="result-line">S symmetric, K skew-symmetric</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 4</span><span class="example-title">AAᵀ is Always Symmetric</span></div>
<div class="example-body">
<p>A = [[1,2],[3,4]]. Verify AAᵀ is symmetric.</p>
<div class="math-block">Aᵀ = [[1,3],[2,4]]
AAᵀ = [[1,2],[3,4]]·[[1,3],[2,4]]
= [[5,11],[11,25]]
(AAᵀ)ᵀ = [[5,11],[11,25]] = AAᵀ ✓</div>
<div class="result-line">AAᵀ is always symmetric</div>
</div>
</div>
</div>
</section>
<!-- ── SYSTEM OF EQUATIONS ── -->
<section class="topic-section" id="mat-system">
<div class="section-header">
<div class="section-number">00h</div>
<h2 class="section-title">System of <span class="highlight">Linear Equations</span></h2>
</div>
<div class="def-box">
<span class="def-label">Definition</span>
A system of m linear equations in n unknowns can be written in matrix form as <strong>AX = B</strong>, where A is the <strong>coefficient matrix</strong> (m×n), X is the <strong>variable column vector</strong> (n×1), and B is the <strong>constant column vector</strong> (m×1). The system has a unique solution if det(A) ≠ 0. Solutions are found via Cramer's Rule, matrix inversion, or Gaussian Elimination.
</div>
<div class="formula-box">
<span class="formula-label">Solution Methods</span>
<div class="math-block">Matrix form: AX = B
Method 1 — Matrix Inversion (if det(A)≠0):
X = A⁻¹B
Method 2 — Cramer's Rule (n×n system):
xᵢ = det(Aᵢ)/det(A)
Aᵢ = A with column i replaced by B
Method 3 — Gaussian Elimination:
Augment [A|B], row-reduce to [I|X]
Consistency (Rouché–Capelli Theorem):
rank(A) = rank([A|B]) → Consistent
rank(A) < rank([A|B]) → Inconsistent
rank(A) = rank([A|B]) = n → Unique solution
rank(A) = rank([A|B]) < n → Infinite solutions</div>
</div>
<div class="examples-grid">
<div class="example-card">
<div class="example-header"><span class="example-num">EX 1</span><span class="example-title">Matrix Inversion Method</span></div>
<div class="example-body">
<p>Solve: 2x+y=5, x+3y=10 → AX=B</p>
<div class="math-block">A=[[2,1],[1,3]], B=[[5],[10]]
det(A) = 6−1 = 5
A⁻¹=(1/5)[[3,−1],[−1,2]]
X = A⁻¹B
x = (1/5)(15−10) = 1
y = (1/5)(−5+20) = 3</div>
<div class="result-line">x = 1, y = 3</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 2</span><span class="example-title">Cramer's Rule</span></div>
<div class="example-body">
<p>Solve: x+2y=8, 3x+y=9</p>
<div class="math-block">D = |1,2;3,1| = 1−6 = −5
Dx = |8,2;9,1| = 8−18 = −10
Dy = |1,8;3,9| = 9−24 = −15
x = −10/−5 = 2
y = −15/−5 = 3</div>
<div class="result-line">x = 2, y = 3</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 3</span><span class="example-title">Gaussian Elimination</span></div>
<div class="example-body">
<p>x+2y+z=6, 2x+y+z=5, x+y+2z=7</p>
<div class="math-block">[A|B]:
[1,2,1|6]
[2,1,1|5] R2−2R1→[0,−3,−1|−7]
[1,1,2|7] R3−R1→[0,−1,1|1]
Back-substitute: z=2, y=1, x=2</div>
<div class="result-line">x=2, y=1, z=2</div>
</div>
</div>
<div class="example-card">
<div class="example-header"><span class="example-num">EX 4</span><span class="example-title">Consistency Check</span></div>
<div class="example-body">
<p>x+y=2, 2x+2y=5 — consistent?</p>
<div class="math-block">A=[[1,1],[2,2]], B=[[2],[5]]
rank(A): R2−2R1=[0,0] → rank=1
[A|B]: R2−2R1=[0,0|1] → rank=2
rank(A)≠rank([A|B])</div>
<div class="result-line">Inconsistent — No solution</div>
</div>
</div>
</div>
</section>
</div><!-- /chapter-group matrix fundamentals -->
<!-- ══════════════════════════════════════
CHAPTER 1: CARLEY HAMINGTON
══════════════════════════════════════ -->
<div class="chapter-group">
<div class="chapter-banner">
<div class="chapter-icon">⬛</div>
<div>
<div class="chapter-name">Chapter 1 — Matrix Theory</div>
<div class="chapter-desc">Cayley-Hamilton Theorem, Characteristic Equations, Eigenvalues & Eigenvectors</div>
</div>
</div>
<!-- Carley Hamington -->
<section class="topic-section" id="ch-carley">
<div class="section-header">
<div class="section-number">01</div>
<h2 class="section-title"><span class="highlight">Cayley-Hamilton</span> Theorem</h2>
</div>
<div class="def-box">
<span class="def-label">Definition</span>
The <strong>Cayley-Hamilton Theorem</strong> states that every square matrix satisfies its own characteristic equation. Named after Arthur Cayley and William Rowan Hamilton, the theorem guarantees that if p(λ) is the characteristic polynomial of A, then <strong>p(A) = 0</strong> — the zero matrix. This is one of the most important results in linear algebra.
</div>
<div class="concept-card">
<div class="concept-label">Concept</div>
<div class="concept-title">Cayley-Hamilton Theorem</div>
<div class="concept-desc">
Every square matrix satisfies its own characteristic equation. If <span class="formula">p(λ) = det(A − λI) = 0</span> is the characteristic polynomial of matrix A, then substituting A for λ gives <span class="formula">p(A) = 0</span> (the zero matrix). This theorem is fundamental for computing matrix inverses and powers.
</div>
</div>
<div class="formula-box">
<span class="formula-label">Formulas</span>
<div class="math-block">Characteristic polynomial: p(λ) = det(A − λI)
For 2×2 matrix: p(λ) = λ² − tr(A)λ + det(A)
Cayley-Hamilton: p(A) = A² − tr(A)A + det(A)I = 0
Finding inverse via C-H:
p(A) = 0 → multiply by A⁻¹
→ A⁻¹ expressed as polynomial in A
Finding Aⁿ (reduce using p(A)=0):
Any Aⁿ = αA + βI for some scalars α, β</div>
</div>
<div class="info-box">
<strong>Key Insight:</strong> You never need to compute high powers of a matrix directly. Using the characteristic polynomial, any power A<sup>n</sup> can be reduced to a linear combination of lower powers.
</div>
<ul class="prop-list">
<li>Applies to all square matrices (n × n), over any field.</li>
<li>Can be used to find the inverse: express I from the polynomial and solve for A⁻¹.</li>
<li>Reduces computation of matrix powers to polynomial arithmetic.</li>
</ul>
<div class="examples-grid">
<div class="example-card">
<div class="example-header">
<span class="example-num">EX 1</span>
<span class="example-title">Verify Cayley-Hamilton for 2×2</span>
</div>
<div class="example-body">
<p>Let A = [[2, 1],[1, 3]]. Find characteristic polynomial and verify A satisfies it.</p>
<div class="math-block">det(A − λI) = (2−λ)(3−λ) − 1
= λ² − 5λ + 5
A² = [[5,5],[5,10]]
A²−5A+5I = [[5,5],[5,10]]
−[[10,5],[5,15]]
+[[5,0],[0,5]]
= [[0,0],[0,0]] ✓</div>
<div class="result-line">Characteristic eqn satisfied by A</div>
</div>
</div>
<div class="example-card">
<div class="example-header">
<span class="example-num">EX 2</span>
<span class="example-title">Find Inverse Using C-H Theorem</span>
</div>
<div class="example-body">
<p>For A = [[1,2],[2,3]], characteristic eqn: λ²−4λ−1 = 0.</p>
<div class="math-block">A² − 4A − I = 0
Multiply by A⁻¹:
A − 4I − A⁻¹ = 0
A⁻¹ = A − 4I
= [[1,2],[2,3]] − [[4,0],[0,4]]
= [[-3,2],[2,-1]]</div>
<div class="result-line">A⁻¹ = [[-3,2],[2,-1]]</div>
</div>
</div>
<div class="example-card">
<div class="example-header">
<span class="example-num">EX 3</span>
<span class="example-title">Compute A⁴ via Polynomial</span>
</div>
<div class="example-body">
<p>For A with char. poly λ²−5λ+6=0, so A²=5A−6I. Then:</p>
<div class="math-block">A³ = A·A² = A(5A−6I)
= 5A²−6A
= 5(5A−6I)−6A
= 19A−30I
A⁴ = A·A³ = A(19A−30I)
= 19A²−30A
= 65A−114I</div>
<div class="result-line">Higher powers simplified without brute-force</div>
</div>
</div>
<div class="example-card">
<div class="example-header">
<span class="example-num">EX 4</span>
<span class="example-title">3×3 Matrix — C-H Verification</span>
</div>
<div class="example-body">
<p>For A = diag(1,2,3), char poly is (λ−1)(λ−2)(λ−3) = λ³−6λ²+11λ−6 = 0.</p>
<div class="math-block">Substituting A:
A³−6A²+11A−6I
= diag(1,8,27)−6·diag(1,4,9)
+11·diag(1,2,3)−6I
= diag(1−6+11−6,
8−24+22−6,
27−54+33−6)
= diag(0,0,0) = 0 ✓</div>
</div>
</div>
</div>
</section>
<!-- Characteristic -->
<section class="topic-section" id="ch-characteristic">
<div class="section-header">
<div class="section-number">02</div>
<h2 class="section-title"><span class="highlight">Characteristic</span> Equation & Eigenvalues</h2>
</div>
<div class="def-box">
<span class="def-label">Definition</span>
The <strong>characteristic equation</strong> of a square matrix A is <strong>det(A − λI) = 0</strong>. The roots of this polynomial are called <strong>eigenvalues</strong> (or characteristic values) of A. For each eigenvalue λᵢ, a non-zero vector <strong>v</strong> satisfying <strong>Av = λᵢv</strong> is called an <strong>eigenvector</strong>. Eigenvalues and eigenvectors describe the intrinsic scaling/rotation properties of a linear transformation.
</div>
<div class="concept-card">
<div class="concept-label">Concept</div>
<div class="concept-title">Characteristic Equation</div>
<div class="concept-desc">
The characteristic equation of a matrix A is <span class="formula">det(A − λI) = 0</span>, where λ is the eigenvalue. Solving this equation gives the eigenvalues of the matrix. For each eigenvalue λᵢ, the eigenvector v is found from <span class="formula">(A − λᵢI)v = 0</span>.
</div>
</div>
<div class="formula-box">
<span class="formula-label">Formulas</span>
<div class="math-block">Characteristic equation: det(A − λI) = 0
Eigenvector equation: (A − λᵢI)v = 0
For 2×2: λ² − (a+d)λ + (ad−bc) = 0
i.e. λ² − tr(A)λ + det(A) = 0
Key relations:
Sum of eigenvalues = tr(A) = a₁₁+a₂₂+...+aₙₙ
Product of eigenvalues = det(A)
Eigenvalues of Aⁿ = λ₁ⁿ, λ₂ⁿ, ...
Eigenvalues of A⁻¹ = 1/λ₁, 1/λ₂, ...</div>
</div>
<div class="info-box">
<strong>Eigenvalue Properties:</strong> The sum of eigenvalues equals the trace of A (sum of diagonal elements), and the product of eigenvalues equals det(A).
</div>
<div class="examples-grid">
<div class="example-card">
<div class="example-header">
<span class="example-num">EX 1</span>
<span class="example-title">2×2 Eigenvalues</span>
</div>
<div class="example-body">
<p>Find eigenvalues of A = [[4,1],[2,3]].</p>
<div class="math-block">det(A−λI) = (4−λ)(3−λ)−2
= λ²−7λ+10 = 0
= (λ−5)(λ−2) = 0</div>
<div class="result-line">λ₁ = 5, λ₂ = 2</div>
</div>
</div>
<div class="example-card">
<div class="example-header">
<span class="example-num">EX 2</span>
<span class="example-title">Find Eigenvectors</span>
</div>
<div class="example-body">
<p>For A = [[4,1],[2,3]], find eigenvectors for λ₁=5.</p>
<div class="math-block">(A−5I)v = 0
[[-1,1],[2,-2]]·[x,y]ᵀ = 0
−x + y = 0 → y = x</div>
<div class="result-line">v₁ = [1, 1]ᵀ (for λ=5)</div>
</div>
</div>
<div class="example-card">
<div class="example-header">
<span class="example-num">EX 3</span>
<span class="example-title">3×3 Characteristic Polynomial</span>
</div>
<div class="example-body">
<p>Find char. eqn of A = [[2,0,0],[1,3,0],[0,1,4]].</p>
<div class="math-block">det(A−λI) = (2−λ)(3−λ)(4−λ) = 0
(upper triangular: eigenvalues
are the diagonal entries)</div>
<div class="result-line">λ = 2, 3, 4</div>
</div>
</div>
<div class="example-card">
<div class="example-header">
<span class="example-num">EX 4</span>
<span class="example-title">Verify Trace & Det</span>
</div>
<div class="example-body">
<p>For A = [[3,1],[0,2]], eigenvalues are λ=3,2.</p>
<div class="math-block">Trace(A) = 3+2 = 5 ✓ (λ₁+λ₂=5)
det(A) = 6 ✓ (λ₁·λ₂=6)</div>
<div class="result-line">Properties verified</div>
</div>
</div>
</div>
</section>
</div>
<!-- ══════════════════════════════════════
CHAPTER 2: VECTOR OVERVIEW
══════════════════════════════════════ -->
<div class="chapter-group">
<div class="chapter-banner">
<div class="chapter-icon">→</div>
<div>
<div class="chapter-name">Chapter 2 — Vector Algebra</div>
<div class="chapter-desc">Multiplication, Angles, Projections and Differentiation</div>
</div>
</div>
<!-- Vector -->
<section class="topic-section" id="ch-vector">
<div class="section-header">