Day-6 Real Analysis & Linear Algebra I Concept To Questions I IFAS

Linear Algebra PYQs in Mathematical Science explained with clear concepts and exam-focused approach. Learn block diagonal matrices, characteristic polynomial, minimal polynomial, eigenspace dimensions, linear operators, trace, rank-nullity, eigenvalues of block matrices 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] Intro [00:04] Q1: 7x7 Block Diagonal Matrix [01:50] Characteristic Polynomial [02:16] Minimal Polynomial from Jordan Block [02:52] Eigenspace Dimensions [03:24] Q1 Answer: First and Fourth [03:36] Q2: Linear Operator T on R³ [04:04] g(T)=0 Analysis [06:55] Two Independent Eigenvectors for 0 [07:14] Q2 Answer: Fourth Option [12:18] Q3: 3x3 Real Matrix False Assertion [13:09] Real Eigenvalue Always Exists [14:27] Determinant Negative Case [15:16] Determinant Positive Need Not All Real [16:08] Q3 Answer: Fourth [16:21] Q4: PQ = -QP for Invertible Matrices [17:31] Similarity: Q Similar to -Q [18:55] Q4 Answer: Trace P = Trace Q = 0 [19:21] Q5: m×n Matrix Rank r, AX=b [20:11] Solution for Each b Implies Onto [21:43] First Option True: m = r [21:49] Second Option False: Column Space Whole R^m [22:14] Third Option False: Null Space Trivial When m = n [23:02] Fourth Option True: m ≥ n Implies m = n [23:33] Q5 Answer: First and Fourth [24:41] Q6: Block Matrix C = [[A,B],[B,A]] [26:18] Counterexample: All Eigenvalues Real False [28:13] Eigenvalue of A+B Implies Eigenvalue of C [29:38] Eigenvalue of A-B Implies Eigenvalue of C [30:26] Eigenvalue of A or B Need Not Be of C [31:18] Q6 Answer: First and Second [31:57] Q7: Char Poly Divisible by t² [32:57] Char Poly x²(x - α) [33:29] Eigenspace for 0 Need Not Be 2D [33:43] All Eigenvalues Real True [33:55] A Need Not Be Nilpotent [34:07] A Need Not Be Diagonalizable [34:26] Q7 Answer: Second Only [34:42] Linear Algebra Session End [35:15] End Of Lecture ━━━━━━━━━━━━━━━━━━━━━━━━━ 📖 Topics Covered in This Lecture: Block diagonal matrix, characteristic polynomial, minimal polynomial, eigenspace dimension, linear operator, trace, rank-nullity, system of linear equations, block matrices eigenvalues, nilpotent matrix, diagonalizability, PYQ solutions. ━━━━━━━━━━━━━━━━━━━━━━━━━ 📚 Lecture Summary: In this session, IFAS Expert Faculty focuses on Linear Algebra PYQs for CSIR NET, GATE and SET. Solved 7 questions covering 7x7 block diagonal complex matrix characteristic and minimal polynomial, eigenspace dimensions; linear operator on R³ with g(T)=0; 3x3 real matrix assertions; invertible matrices PQ=-QP trace; m×n matrix AX=b solution; block matrix C=[[A,B],[B,A]] eigenvalues; and 3x3 matrix with characteristic polynomial divisible by t². Learn shortcuts, counterexamples 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