Eigen Values of AB and BA | LINEAR ALGEBRA | CSIR NET | IFAS

Eigenvalues of AB and BA in Linear Algebra explained with clear concepts and exam-focused approach. Learn non-zero eigenvalues, characteristic polynomial, minimal polynomial, diagonalizability, rank and PYQ tricks for CSIR NET, GATE and SET. 🧭 Get Personalized Preparation Guidance: https://ifasonline.com/csir-net/mathe... 🔥Join India's Best Coaching for CSIR NET and GATE Exams: https://ifasonline.com/csir-net/mathe... 📞 Need Guidance? Call/WhatsApp: 9172266888 ━━━━━━━━━━━━━━━━━━━━━━━━━ ⏳ Lecture Timeline: [00:00] Introduction [00:28] Batch and Test Series Info [01:33] Concept: Eigenvalues of AB and BA [02:17] Result: Non-zero Eigenvalues Same [02:32] Proof: ABv = λv then BA(Bv) = λ(Bv) [03:54] Important: Algebraic and Geometric Multiplicity Same [04:42] Characteristic Polynomial: AB and BA [05:31] Formula: char poly AB = x^(m-n) times char poly BA [06:58] Special Case: m = n [07:51] Minimal Polynomial of AB and BA [08:24] Example: A and B Matrices [09:26] Diagonalizability: AB vs BA [10:18] Condition for BA Diagonalizable [12:17] Question 1: Real Symmetric Matrices AB [13:35] Solution: Eigenvalues of BA [14:40] Minimal Polynomial Same, BA Not Diagonalizable [15:23] Question 2: A 5x3, B 3x5, C = AB [15:47] At Least Two Zero Eigenvalues [16:10] Geometric Multiplicity Same for Nonzero [16:33] Rank of C and D Not Equal [17:20] Correct Options A and B [17:32] Homework [18:12] End Of Lecture ━━━━━━━━━━━━━━━━━━━━━━━━━ 📖 Topics Covered in This Lecture: Eigenvalues of AB and BA, non-zero eigenvalues, algebraic multiplicity, geometric multiplicity, characteristic polynomial, minimal polynomial, diagonalizability, rank, PYQ solutions. ━━━━━━━━━━━━━━━━━━━━━━━━━ 📚 Lecture Summary: In this session, IFAS Expert Faculty focuses on Eigenvalues of AB and BA for CSIR NET, GATE and SET. Key results: equal non-zero eigenvalues, same algebraic and geometric multiplicity, characteristic polynomial relation, minimal polynomial differs by x factor, and diagonalizability conditions. Solved PYQs on real symmetric matrices and A 5x3, B 3x5 with C=AB. Learn shortcuts and exam-focused problem-solving. ━━━━━━━━━━━━━━━━━━━━━━━━━ 🎓 This lecture is highly beneficial for students preparing for: ✅ CSIR NET Mathematical Science ✅ GATE Mathematics ✅ SET Mathematics ✅ TIFR Mathematics ✅ PhD Entrance Exams ━━━━━━━━━━━━━━━━━━━━━━━━━ 🏆 Why IFAS is India’s No. 1 Choice for CSIR NET & GATE Mathematics? (Our Results Speak): ✅ CSIR NET June 2026: 4500+ selection ( AIR 1,3 ,4 6, 8 ,...) ✅ CSIR NET Dec 2025: 4000+ Selections (AIR 3, 8, 9.....) ✅ CSIR NET June 2025: 4200+ Selections (AIR 1, 2, 6, 10...) ✅ CSIR NET Dec 2024: 4000+ Selections (AIR 5, 6, 7, 8, 9...) ✅ CSIR NET June 2024: 2000+ Selections (AIR 3, 10, 13...) ✅ CSIR NET Dec 2023: 1200+ Selections (AIR 2, 4, 5...) 📄 View IFAS Result Sheet: 👉 https://drive.google.com/file/d/1THh6... ━━━━━━━━━━━━━━━━━━━━━━━━━ 🔗 Important Links: 🌍 Visit our Website: https://ifasonline.com 📞 Contact for Queries: 9172266888 ━━━━━━━━━━━━━━━━━━━━━━━━━ 🔗 India's Best Online Classes and Books: 🔥CSIR NET Mathematical Science Live & Recorded Course: https://ifasonline.com/csir-net/mathe... 🎯 GATE Mathematics Recorded Course: https://ifasonline.com/gate/mathemati... 📘CSIR NET Mathematical Science Books: https://ifasonline.com/csir-net/mathe... ━━━━━━━━━━━━━━━━━━━━━━━━━ 📲 Download IFAS App: Android: https://play.google.com/store/apps/de... IOS: https://apps.apple.com/us/app/ifas/id... ━━━━━━━━━━━━━━━━━━━━━━━━━ Stay Updated Via Other Social Media: ✔Telegram: https://t.me/NET_GATE_SET_Maths_Discu... ✔Instagram:   / csir_net_mathematical_science   #csirnetmathematicalscience #gatemathematics #csirnetmaths #csirnetdec2026 #csirnetlecture #ifasmathematics #linearalgebra #eigenvalues