GATE Engineering Mathematics Linear Algebra - Quick Notes

Matrix Algebra

Matrix Operations - Basics

Matrix Addition & Subtraction

\((A \pm B)_{ij} = A_{ij} \pm B_{ij}\) (same dimensions required)

Matrix Multiplication

\((AB)_{ij} = \sum_{k=1}^{n} A_{ik}B_{kj}\)

  • \(A_{m \times n} \cdot B_{n \times p} = C_{m \times p}\)

  • Generally \(AB \neq BA\)

Transpose

\((A^T)_{ij} = A_{ji}\)

  • \((A^T)^T = A\)

  • \((AB)^T = B^T A^T\)

Special Matrices

Identity Matrix

\(I_n = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}\)

\(AI = IA = A\)

Symmetric Matrix

\(A = A^T\)

Diagonal Matrix

\(A_{ij} = 0\) for \(i \neq j\)

Orthogonal Matrix

\(AA^T = A^T A = I\)

  • \(A^{-1} = A^T\)

  • \(|A| = \pm 1\)

Matrix Inverse

Definition

\(A^{-1}\) exists if \(AA^{-1} = A^{-1}A = I\)

Properties

  • \((A^{-1})^{-1} = A\)

  • \((AB)^{-1} = B^{-1}A^{-1}\)

  • \((A^T)^{-1} = (A^{-1})^T\)

  • \(A^{-1}\) exists \(\Leftrightarrow\) \(\det(A) \neq 0\)

For 2×2 Matrix

\(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\)

\(A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\)

Determinants

Properties

  • \(\det(AB) = \det(A)\det(B)\)

  • \(\det(A^T) = \det(A)\)

  • \(\det(A^{-1}) = \frac{1}{\det(A)}\)

  • If any row/column is zero, \(\det(A) = 0\)

  • Swapping rows changes sign of determinant

Calculation Methods

  • Cofactor expansion

  • Row/Column operations

  • For triangular matrix: product of diagonal elements

Matrix Rank

Definition

Rank of matrix = Number of linearly independent rows (or columns)

Properties

  • \(\text{rank}(A) \leq \min(m,n)\) for \(A_{m \times n}\)

  • \(\text{rank}(A) = \text{rank}(A^T)\)

  • \(\text{rank}(AB) \leq \min(\text{rank}(A), \text{rank}(B))\)

  • Full rank: \(\text{rank}(A) = \min(m,n)\)

Finding Rank

  1. Convert to Row Echelon Form (REF)

  2. Count non-zero rows

Systems of Linear Equations

System of Linear Equations

General Form

\(Ax = b\) where \(A_{m \times n}\), \(x_{n \times 1}\), \(b_{m \times 1}\)

Augmented Matrix

\([A|b] = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} & | & b_1 \\ a_{21} & a_{22} & \cdots & a_{2n} & | & b_2 \\ \vdots & \vdots & \ddots & \vdots & | & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & | & b_m \end{pmatrix}\)

Solution Methods

  • Gaussian Elimination

  • Gauss-Jordan Method

  • Matrix Inversion (if \(A\) is square and invertible)

  • Cramer’s Rule (for square systems)

Solution Types

Let \(r(A) =\) rank of \(A\), \(r(A|b) =\) rank of augmented matrix

Consistent System

\(r(A) = r(A|b)\)

  • Unique Solution: \(r(A) = r(A|b) = n\)

  • Infinite Solutions: \(r(A) = r(A|b) < n\)

Inconsistent System

\(r(A) \neq r(A|b)\) \(\Rightarrow\) No solution

Homogeneous System (\(Ax = 0\))

  • Always consistent (trivial solution \(x = 0\))

  • Non-trivial solution exists \(\Leftrightarrow\) \(r(A) < n\)

Cramer’s Rule

For Square System \(Ax = b\) with \(\det(A) \neq 0\)

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

where \(A_i\) is matrix \(A\) with \(i\)-th column replaced by \(b\)

Example for 2×2

\(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} p \\ q \end{pmatrix}\)

\(x = \frac{\begin{vmatrix} p & b \\ q & d \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}\), \(y = \frac{\begin{vmatrix} a & p \\ c & q \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}\)

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors

Definition

For square matrix \(A\):

\[Av = \lambda v\]
where \(v \neq 0\) (eigenvector), \(\lambda\) (eigenvalue)

Characteristic Equation

\[\det(A - \lambda I) = 0\]
This gives the characteristic polynomial

Steps to Find Eigenvalues & Eigenvectors

  1. Solve \(\det(A - \lambda I) = 0\) for \(\lambda\)

  2. For each \(\lambda_i\), solve \((A - \lambda_i I)v = 0\) for \(v\)

Properties of Eigenvalues

Important Properties

  • Sum of eigenvalues = Trace of \(A\) = \(\sum a_{ii}\)

  • Product of eigenvalues = \(\det(A)\)

  • Eigenvalues of \(A^T\) = Eigenvalues of \(A\)

  • Eigenvalues of \(A^{-1}\) = \(\frac{1}{\lambda_i}\) (if \(A\) invertible)

  • Eigenvalues of \(A^k\) = \(\lambda_i^k\)

For Symmetric Matrices

  • All eigenvalues are real

  • Eigenvectors are orthogonal

  • Always diagonalizable

Diagonalization

Definition

Matrix \(A\) is diagonalizable if \(A = PDP^{-1}\) where \(D\) is diagonal

Condition for Diagonalization

\(A\) is diagonalizable \(\Leftrightarrow\) \(A\) has \(n\) linearly independent eigenvectors

Diagonalization Process

  1. Find eigenvalues \(\lambda_1, \lambda_2, \ldots, \lambda_n\)

  2. Find corresponding eigenvectors \(v_1, v_2, \ldots, v_n\)

  3. Form \(P = [v_1 | v_2 | \cdots | v_n]\)

  4. Form \(D = \text{diag}(\lambda_1, \lambda_2, \ldots, \lambda_n)\)

  5. Then \(A = PDP^{-1}\)

Quick Reference Formulas

Key Formulas for GATE

  • Determinant 2×2: \(ad - bc\)

  • Determinant 3×3: Use cofactor expansion

  • Inverse 2×2: \(\frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\)

  • Characteristic polynomial: \(\det(A - \lambda I) = 0\)

  • Trace = Sum of diagonal elements = Sum of eigenvalues

Common GATE Topics

  • Rank calculation using row operations

  • Solution types of linear systems

  • Eigenvalue problems for 2×2 and 3×3 matrices

  • Properties of symmetric matrices

  • Orthogonal matrices and their properties

Practice Tips

For GATE Preparation

  • Master 2×2 and 3×3 determinant calculations

  • Practice row operations for finding rank

  • Memorize eigenvalue properties

  • Focus on symmetric matrix properties

  • Practice Cramer’s rule for small systems

  • Understand geometric interpretation of eigenvectors

Time-Saving Tips

  • Use properties of special matrices

  • For triangular matrices: eigenvalues = diagonal elements

  • Check answer using trace and determinant properties

  • Use elimination techniques efficiently