Linear Algebra L3 |Orthogonal, Idempotent, Nilpotent Matrix | GATE DA

What if you could answer a GATE matrix question without writing the matrix down? This is Lecture 3 of the PrepNeuron Linear Algebra course for GATE DA 2027. We cover four special matrices: orthogonal, idempotent, nilpotent and involutory. Each one comes with shortcuts for its determinant, eigenvalues, rank or inverse. Once you spot the type, the answer follows without any calculation. 🧠 Daily Practice Challenge: 10 GATE DA MCQs every day with instant answers and explanations. Join on Telegram 👉 https://t.me/prepneuron_gateda 📌 In this lecture • Orthogonal matrix: AᵀA = I, so A⁻¹ = Aᵀ, det = ±1, lengths preserved • Idempotent matrix: A² = A, eigenvalues 0 or 1, rank = trace • Nilpotent matrix: Aᵏ = O, every eigenvalue 0, det = trace = 0 • Involutory matrix: A² = I, its own inverse, eigenvalues ±1 • Periodic matrix, and how it connects to idempotent • All five side by side: the one-line summary for the exam hall • 3 solved examples + the 4 most common mistakes Solved examples Q1. Idempotent matrix of order 3 with trace 2: find its rank and eigenvalues Q2. Nilpotent matrix of order 4: find det(A) and trace(A) Q3. Verify a matrix is orthogonal and write its inverse in one step Course: Linear Algebra for GATE DA (59 lectures · 8 modules) Module 1: Matrix Foundations (Lectures 1–8) Previous → L2: Types of Matrices (diagonal, triangular, symmetric, skew-symmetric) Next → L4: Matrix Multiplication and Its Properties Full playlist:    • Linear Algebra   Who is this for: GATE DA 2027 aspirants, GATE CSE / ECE students revising Engineering Mathematics, and anyone learning linear algebra for machine learning. Subscribe to PrepNeuron for concept lectures, PYQ solving and revision for GATE DA. Train your thinking. PrepNeuron, an education initiative of NARAYANI. #PrepNeuron #GATEDA #GATE2027 #LinearAlgebra #SpecialMatrices #OrthogonalMatrix #IdempotentMatrix #NilpotentMatrix #GATEDA2027 #LinearAlgebraForGATE #EngineeringMathematics #GATEMaths #DataScience #MachineLearning #GATECSE