Math — Advanced Linear Algebra

A progressive journey from manually computing determinants with Python lists to classifying matrix definiteness with NumPy — building deep intuition for the linear algebra pipeline: determinant → minor → cofactor → adjugate → inverse → definiteness.

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Learning Objectives

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Task-by-Task Reference

Each task below highlights the unique challenge it posed and the new technique introduced — techniques from earlier tasks are not repeated. Tasks 0–4 use only Python lists (no NumPy); Task 5 introduces NumPy.

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Task 0 — Determinant (0-determinant.py)

Challenge: Compute the determinant of a square matrix of arbitrary size using only Python lists — implementing Laplace expansion from scratch with proper input validation.

Approach: The function handles three base cases (0×0 → 1, 1×1 → the value, 2×2 → $ad - bc$) and recursively expands along the first row for larger matrices. For each column $j$, it builds the submatrix by skipping row 0 and column $j$, then accumulates $(-1)^j \cdot a_{0j} \cdot \det(\text{submatrix})$.

New techniques introduced:

Key takeaway: The determinant is the foundation of the entire linear algebra pipeline. Laplace expansion recursively breaks an n×n problem into n problems of size (n−1)×(n−1) — pure divide and conquer.

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Task 1 — Matrix of Minors (1-minor.py)

Challenge: Build a full matrix of minors — for every position $(i, j)$, compute the determinant of the submatrix formed by deleting row $i$ and column $j$.

Approach: Reuses determinant() from Task 0. Implements special-case shortcuts for 1×1 ([[1]]) and 2×2 (reverse rows and columns). For 3×3+, iterates over all $(i, j)$ pairs, builds each submatrix by skipping row $i$ and column $j$, computes its determinant, then reassembles results into the minor matrix shape using list slicing temp[i*n:(i+1)*n].

New techniques introduced:

Key takeaway: The minor matrix generalizes the determinant — instead of one number, you compute a determinant for every position $(i, j)$. Each minor tells you "how much does this position matter?"

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Task 2 — Cofactor Matrix (2-cofactor.py)

Challenge: Convert the matrix of minors into a cofactor matrix by applying the checkerboard sign pattern $(-1)^{i+j}$.

Approach: Calls minor() from Task 1, then iterates over every element multiplying by $(-1)^{\text{row}+\text{col}}$. The sign pattern creates alternating signs like a chess board: $+$ at even $i+j$, $-$ at odd $i+j$.

New techniques introduced:

Key takeaway: The cofactor is just a signed minor — same values, alternating signs. The sign pattern is what turns minors into the building blocks of inverses.

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Task 3 — Adjugate Matrix (3-adjugate.py)

Challenge: Compute the adjugate (classical adjoint) — the transpose of the cofactor matrix. This is the final preprocessing step before the inverse.

Approach: Calls cofactor() from Task 2, then transposes by iterating with cofact[col][row] (swapped indices). Reassembles into 2D using row slicing.

New techniques introduced:

Key takeaway: The adjugate is the transpose of the cofactor matrix — a simple operation that completes the preprocessing chain. The inverse is now just one division away: $A^{-1} = \text{adj}(A) / \det(A)$.

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Task 4 — Inverse (4-inverse.py)

Challenge: Compute the matrix inverse by dividing the adjugate by the determinant — and detect singular (non-invertible) matrices.

Approach: Calls adjugate() and determinant(). If $\det = 0$, returns None (singular). Otherwise divides every element of the adjugate by the determinant: adju[row][col] / det.

New techniques introduced:

Key takeaway: The inverse is the culmination of the entire pipeline — every previous step (determinant, minor, cofactor, adjugate) was building toward this. If $\det = 0$, the matrix is singular and has no inverse.

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Task 5 — Definiteness (5-definiteness.py)

Challenge: Classify a symmetric matrix by its definiteness (positive/negative definite, semi-definite, or indefinite) — introducing NumPy and eigenvalue analysis.

Approach: Validates input as np.ndarray, checks symmetry with np.allclose(matrix, matrix.T), then uses np.linalg.eigvalsh() to get eigenvalues. Classifies by sign: all $> 0$ → positive definite, all $\geq 0$ → positive semi-definite, all $< 0$ → negative definite, all $\leq 0$ → negative semi-definite, mixed → indefinite.

New techniques introduced:

Key takeaway: Definiteness tells you the "shape" of a matrix — eigenvalues are the key. All positive eigenvalues = positive definite (bowl-shaped), mixed signs = indefinite (saddle-shaped). This concept is critical for optimization and ML.

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Bonus — Matrix Quiz (quiz.py)

Challenge: A comprehensive 10-question quiz that tests all six concepts using NumPy's linalg module as a verification tool — applying the theory to concrete 3×3 matrices.

Approach: Implements NumPy-based helper functions (minor_matrix, cofactor_matrix, adjugate_matrix, classify_definiteness) and runs each concept against multiple-choice options using np.allclose(). Also explores eigenvalue power iteration ($A^{10}v = \lambda^{10}v$) and eigenpair verification.

New techniques introduced:

Key takeaway: The quiz validates your manual implementations against NumPy's battle-tested linalg module — and shows how eigenvalues unlock efficient matrix powers without computing $A^{10}$ directly.

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Technique Inventory

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Resources

• NumPy linalg.det — Official Docs
• NumPy linalg.eigvalsh — Official Docs
• NumPy linalg.inv — Official Docs
• Matrix Minor, Cofactor, Adjugate — Wikipedia
• Definiteness of a Matrix — Wikipedia
• Laplace Expansion — Wikipedia