Linear Algebra (Matrices, Eigenvalues)

Syllabus Unit: Engineering Mathematics

Why it matters

Chapter overview

Linear algebra is GATE's most heavily weighted Engineering Mathematics topic, appearing in every branch's paper — matrices encode systems of linear equations, determinants test whether solutions exist and are unique, and eigenvalues/eigenvectors reveal a matrix's natural directions and scaling factors, feeding directly into control systems, structural analysis, and machine learning.

General Aptitude & Mathematics Engineering Mathematics 15% weightage

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Row swap flips the sign of the determinant

Scaling one row by k scales det by k

Identical or proportional rows force det = 0

det(AB) = det(A) · det(B)

1Matrix Operations & Determinants

Matrix multiplication, transpose and the determinant are the arithmetic toolkit every later linear-algebra topic — rank, inverse, eigenvalues — is built from.

Learning Objectives

Multiply two conformable matrices and compute the transpose, stating when AB ≠ BA.
Compute a 2x2 and 3x3 determinant using cofactor expansion.
Apply determinant properties (row swap flips sign, scaling a row scales det, identical rows give det = 0) to simplify calculations.
Classify special matrices — symmetric, skew-symmetric, orthogonal, idempotent — by their defining algebraic property.

Key Terms

Cofactor Cij(−1)^(i+j) times the minor Mij (determinant of the submatrix left after deleting row i, column j); used to expand a determinant.

Minor MijThe determinant of the smaller matrix left after deleting row i and column j from the original matrix.

Symmetric MatrixA square matrix where A = Aᵀ, i.e. aij = aji for all i, j.

Orthogonal MatrixA square matrix where AAᵀ = I, so its inverse equals its transpose: A⁻¹ = Aᵀ.

A matrix is a compact way to represent a linear system or a linear transformation. Multiplication of matrices is defined row-by-column and is generally **not commutative** (AB ≠ BA), while (AB)ᵀ = BᵀAᵀ and (A⁻¹)ᵀ = (Aᵀ)⁻¹ always hold.

Computing determinants

2x2: det[[a,b],[c,d]] = ad − bc.
3x3 (and larger): cofactor expansion along any row or column — det(A) = Σ aij·Cij for a fixed row i.
Triangular matrix (upper or lower): det = product of the diagonal entries.
det(AB) = det(A)·det(B), and det(A⁻¹) = 1/det(A) when A is invertible.

Exam TipAlways expand along the row or column with the most zero entries — it eliminates whole cofactor terms and is the single biggest time-saver on GATE determinant questions.

Special matrix types GATE tests

Symmetric — A = Aᵀ (aij = aji).
Skew-symmetric — Aᵀ = −A, so every diagonal entry is 0 and aij = −aji.
Orthogonal — AAᵀ = I, so A⁻¹ = Aᵀ; rows/columns are unit vectors, mutually perpendicular.
Idempotent — A² = A (e.g. projection matrices).

Common MistakeA skew-symmetric matrix of ODD order always has det(A) = 0. Proof sketch: det(A) = det(Aᵀ) = det(−A) = (−1)ⁿdet(A); for odd n this forces det(A) = −det(A), so det(A) = 0. A very reliable GATE shortcut.