Nilpotent Matrix Property T^n = 0 Explained | CSIR NET & GATE Mathematics

In this video, we solve and analyze an important Linear Algebra question on Nilpotent Matrices ($T^n = 0$) relevant for CSIR NET Mathematical Sciences, GATE (MA), and IIT JAM. 📌 Problem Discussed: Let T be an n × n matrix with the property Tⁿ = 0. Which of the following is/are true? 1. T has n distinct eigenvalues 2. T has one eigenvalue of multiplicity n 3. 0 is an eigenvalue of T 4. T is similar to a diagonal matrix Key concepts covered: • Definition and properties of a Nilpotent Matrix • Characteristic polynomial and Minimal polynomial for Tⁿ = 0 • Eigenvalues and Algebraic Multiplicity of Nilpotent operators • Diagonalizability condition for Nilpotent Matrices ⏱️ Timestamps: 0:00 - Question Overview 0:45 - Concept of Nilpotent Matrix ($T^n = 0$) 1:40 - Finding Eigenvalues & Multiplicity (Options 1, 2 & 3) 3:30 - Diagonalizability & Counter-example (Option 4) 4:50 - Final Answer & Summary If this explanation helped clear your concepts, make sure to Like, Share, and Subscribe for more CSIR NET & GATE Maths practice problems! #CSIRNET #GATEMaths #LinearAlgebra #Mathematics #IITJAM #NilpotentMatrix #HigherMathematics #linearalgebra #csirnet #csirnetmaths #highermathematics #gatemaths #iitjam #jordancanonicalform #csirnet2024 #mathstricks