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−bcad - bcad−bc) and recursively expands along the first row for larger matrices. For each column jjj, it builds the submatrix by skipping row 0 and column jjj, then accumulates (−1)j⋅a0j⋅det⁡(submatrix)(-1)^j \cdot a_{0j} \cdot \det(\text{submatrix})(−1)j⋅a0j​⋅det(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)(i, j)(i,j), compute the determinant of the submatrix formed by deleting row iii and column jjj.

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)(i, j)(i,j) pairs, builds each submatrix by skipping row iii and column jjj, 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)(i, j)(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(-1)^{i+j}(−1)i+j.

Approach: Calls minor() from Task 1, then iterates over every element multiplying by (−1)row+col(-1)^{\text{row}+\text{col}}(−1)row+col. The sign pattern creates alternating signs like a chess board: +++ at even i+ji+ji+j, −-− at odd i+ji+ji+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=adj(A)/det⁡(A)A^{-1} = \text{adj}(A) / \det(A)A−1=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\det = 0det=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\det = 0det=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> 0>0 → positive definite, all ≥0\geq 0≥0 → positive semi-definite, all <0< 0<0 → negative definite, all ≤0\leq 0≤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 (A10v=λ10vA^{10}v = \lambda^{10}vA10v=λ10v) 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 A10A^{10}A10 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