4.5.10 · D5 · HinglishLinear Algebra (Full)
Question bank — Row echelon form and reduced row echelon form
4.5.10 · D5· Maths › Linear Algebra (Full) › Row echelon form and reduced row echelon form
Is bank mein Gaussian Elimination, Gauss-Jordan Elimination, Rank of a Matrix, Free and Basic Variables, Solving Systems of Linear Equations, Linear Independence, Matrix Inverse via RREF aur Elementary Matrices ke ideas hain — lekin har answer definitions se reason kiya gaya hai, memorise nahi kiya.
True or false — justify
Ek matrix REF mein ho sakti hai lekin RREF mein nahi.
True — REF sirf staircase aur pivots ke neeche zeros maangta hai; RREF leading 1's aur pivot ke upar zeros bhi maangta hai, toh wala pivot ya pivot ke upar koi nonzero entry RREF nahi hai.
Kisi bhi given matrix ka RREF unique hota hai.
True — chahe tum kaun si bhi valid row operations ki sequence choose karo, tum hamesha usi ek RREF pe pahunchte ho; pivot positions aur cleaned entries forced hain.
Kisi bhi given matrix ka REF unique hota hai.
False — staircase ka shape (kaun se columns mein pivots hain) forced hai, lekin actual numbers nahi; alag scalings aur replacements alag REFs dete hain, jaise pivots vs .
Agar do matrices ka RREF same hai, toh unka solution set bhi same hai.
True — RREF solution-preserving row operations se pahuncha jaata hai, toh same RREF wali matrices row-equivalent hain aur same system ke solutions describe karti hain.
Har matrix ka ek RREF hota hai.
True — Gauss–Jordan hamesha terminate karta hai: har column ya toh pivot leta hai ya skip ho jaata hai, aur process finite hai, toh RREF hamesha exist karta hai (all-zero matrix bhi, jiska RREF woh khud hai).
Zero matrix already RREF mein hai.
True — uske koi non-zero rows nahi hain, toh conditions 1–5 vacuously hold karti hain (check karne ke liye koi pivot nahi); kuch clean karne ki zaroorat nahi.
Saari zeros ki ek row pivot row count hoti hai.
False — pivot row ka pehla non-zero entry pivot hota hai, aur zero row mein koi nahi hota, toh yeh koi pivot aur rank contribute nahi karta.
Ek row ka khud se multiple add karna (yaani ) ek legal elementary row operation hai.
False — legal "replace" op hai jisme ; same row use karna use se multiply karta hai, jo actually ek scale hai aur reversible tabhi hai jab — ise galat label karna accidentally ek row ko zero karne ki jagah de sakta hai.
Ek row ko se scale karna ek allowed operation hai.
False — scaling mein chahiye; se multiply karna ek equation ko bana deta hai aur reversible nahi hai, toh yeh solution information destroy kar sakta hai.
Agar kisi matrix mein columns se zyada rows hain, toh woh RREF mein nahi ho sakti.
False — extra rows simply bottom mein zero rows ban jaati hain; RREF kisi bhi shape ko handle karta hai, tall ho ya wide.
Ek augmented matrix ke last column mein pivot hona matlab system ka koi solution nahi hai.
True — woh pivot ek aise row ko represent karta hai jaise , ek impossible equation, toh system inconsistent hai.
Ek coefficient (unaugmented) matrix ke last column mein pivot hona matlab koi solution nahi hai.
False — ek plain coefficient matrix mein last column sirf ek aur variable ka column hai; wahan ek pivot perfectly fine hai aur simply us variable ko basic banata hai.
Row-equivalent matrices ka rank hamesha equal hota hai.
True — rank pivot ki count ke equal hoti hai, aur row operations pivot count nahi badlti, toh equivalent matrices apna rank share karti hain.
Spot the error
"Maine ek staircase banaya jisme har pivot ke neeche zeros hain, toh yeh RREF hai."
Error — yeh sirf REF hai; RREF mein bhi har pivot ke equal hona chahiye aur har pivot ke upar zeros hone chahiye, jo tumne abhi nahi kiya.
"Mere REF mein ek pivot hai, toh maine kahin galti ki hai."
Koi galti nahi — REF mein koi bhi non-zero pivot value allowed hai; sirf RREF pivots ko exactly force karta hai.
"Column 3 mein ek pivot hai aur column 1 mein do rows neeche aur left mein ek nonzero number hai, toh staircase theek hai."
Error — staircase mein har pivot strictly upar wale se right mein hona chahiye; ek pivot ke left-and-below koi nonzero entry matlab pehla pivot skip ho gaya aur form REF nahi hai.
"Matrix mein 3 variables aur rank 3 hai, toh iske infinitely many solutions hone chahiye."
Error — variables ki count ke equal full rank (consistent system ke saath) ek unique solution deta hai, infinitely many nahi; free variables tabhi aate hain jab rank variable count se kam ho.
"Mujhe ek zero row mili, toh system inconsistent hai."
Error — ek zero row () ek redundant, hamesha-sach equation hai, contradiction nahi; inconsistency ke liye augmented column mein ek pivot chahiye ().
"Dono matrices same staircase shape pe reduce hue, toh woh same matrix hain."
Error — same pivot positions matlab same rank aur same pivot columns, lekin actual entries aur hence dono matrices alag ho sakti hain; sirf identical RREFs identical row spaces force karte hain.
"Rank count karne ke liye maine har row ko count kiya jo sabse bottom row nahi thi."
Error — rank REF mein non-zero rows count karta hai (equivalently, pivots ki count), position se independent; bottom rows non-zero ho sakti hain aur top rows theoretically sorting se pehle zero ho sakti hain.
Why questions
REF mein saari all-zero rows bottom pe kyun honi chahiye?
Kyunki pivots ko down-and-right march karna hota hai; beech mein ek zero row pivot columns ki strictly-descending staircase tod deti.
RREF mein pivots ke upar entries clear kyun karte hain lekin plain REF mein nahi?
Neeche clear karna back-substitution ke liye variables ko isolate karne ke liye kaafi hai (REF ka kaam); upar bhi clear karne se har variable exactly ek equation mein aata hai, toh answer bina kisi substitution ke read off kiya ja sakta hai.
Teen row operations kabhi koi solution gain ya lose kyun nahi karte?
Har ek reversible hai — tum ek swap undo kar sakte ho, se un-scale kar sakte ho, ya added multiple wapas subtract kar sakte ho — aur ek reversible transformation solution set ko exactly apne upar map karta hai.
Variables se kam pivots hona (jab consistent ho) infinitely many solutions kyun deta hai?
Har pivot-free column ek free variable hai jise tum koi bhi value de sakte ho, aur aisa har choice ek valid solution deta hai, toh solution set ek poora line/plane hai rather than ek point.
Original matrix ke pivot columns linearly independent kyun hote hain?
Row operations columns ke beech linear dependence relations preserve karti hain, aur RREF mein pivot columns distinct standard basis vectors hote hain (sirf entry ek ), jo clearly independent hain — dekho Linear Independence.
Gauss–Jordan ko matrix inverse compute karne ke liye kyun use kiya ja sakta hai?
augment karna aur ko pe reduce karna exact inverse operations pe apply karta hai, toh right block ban jaata hai; yeh sirf tab kaam karta hai jab fully pe reduce ho (yaani invertible ho).
Hum sirf divide karne ki bajay pivot position mein ek non-zero entry swap kyun karte hain?
Agar current pivot slot hai toh tum use divide nahi kar sakte, toh tum ek aise row ko upar swap karte ho jisme wahan ek non-zero entry ho — swap zero se division avoid karta hai aur staircase ko form karte rehta hai.
Edge cases
zero matrix ka RREF kya hai?
Woh zero matrix khud hai — koi pivot exist nahi karta, toh saari conditions vacuously hold karti hain aur kuch nahi badlta.
jaise single-row matrix ke liye REF kaisa dikhta hai?
Yeh already REF mein hai (column 3 mein ek pivot, neeche koi rows nahi); leading ke liye scale karna ise additionally RREF banata.
Agar ek coefficient matrix identity pe reduce ho jaata hai, toh system ke kitne solutions hain?
Exactly ek — teen variables ke liye teen pivots koi free variables nahi chodta, toh solution augmented column se read kiya gaya unique vector hai.
Ek consistent system ka rank hai aur variables hain; solution set kitna bada hai?
free variables hain, toh solutions ek 2-dimensional family (plane ki tarah) form karti hain, infinitely many.
Kya ek inconsistent system ka phir bhi ek well-defined REF/RREF ho sakta hai?
Haan — reduction hamesha complete hoti hai; inconsistency simply augmented column mein ek pivot ke roop mein dikhti hai (ek row), aur RREF phir bhi unique hai.
Ek matrix jiska RREF mein non-zero rows hain, uski rank kya hai?
Rank hai, kyunki rank pivots ki count ke equal hai jo kisi bhi REF/RREF mein non-zero rows ki count ke equal hai — se bounded.
Kya ek nonzero row wali matrix RREF mein hai?
Nahi — pivot hai, nahi; se scale karke lena ise RREF banata hai (entry upar/saath theek hai kyunki woh pivot column mein nahi hai).
Recall Traps ki ek-line summary
REF ek staircase hai (unique nahi, koi bhi pivot value); RREF upar bhi clean karta hai (unique, leading 1's); zero rows aur pivots-in-the-augmented-column decide karte hain redundant vs impossible; aur rank = pivots = non-zero rows hamesha.