Gaussian elimination — forward elimination, back substitution: Question bank

4.5.9 · D5 · Maths › Linear Algebra (Full) › Gaussian elimination — forward elimination, back substitution

True or false — justify

Row operations system ka solution set change kar dete hain.

False — teeno elementary operations solution set ko exactly preserve karti hain; isliye ye hi only legal moves hain.

Kisi row ko

0

se multiply karna ek valid elementary row operation hai.

False — scaling nonzero constant se hi honi chahiye;

0

se scale karna equation ko erase kar deta hai aur solutions destroy kar sakta hai (ye reversible nahi hai).

Do rows ko swap karna change kar sakta hai ki solutions

x_{1}, \dots, x_{n}

kya hain.

False — swap sirf equations ko reorder karta hai; wahi

x

unhe satisfy karta rehta hai, isliye solution set untouched rehta hai.

Forward elimination hamesha ek upper-triangular matrix produce karta hai jisme har diagonal entry nonzero ho.

False — ye upper-triangular shape produce karta hai, lekin ek diagonal (pivot position) zero bhi ho sakti hai, jo free variable signal karta hai ya column skip karne ko majboor karta hai.

[00 \dots 0 ∣ 0]

form ki row ka matlab hai system ka koi solution nahi hai.

False — all-zero row (RHS sameit) ek redundant equation (

0 = 0

) hai, jo free variable aur hence infinitely many solutions allow karta hai.

[00 \dots 0 ∣ 5]

form ki row ka matlab hai system ka koi solution nahi hai.

True — ye

0 = 5

assert karta hai, jo ek contradiction hai, isliye system baaki rows se independent inconsistent hai.

Agar coefficient matrix already upper-triangular hai, to forward elimination ko kuch nahi karna.

True — diagonal ke niche har entry already zero hai, isliye koi multiplier ki zaroorat nahi; seedha back substitution pe jaao.

Back substitution top row se shuru ki ja sakti hai.

False — sirf bottom row hi ek single unknown isolate karti hai; upar se shuru karne par woh row abhi bhi kai unknowns mix karti rehti hai jo tumhe abhi pata nahi.

Multiplier

m_{ik} = a_{ik} / a_{k k}

negative ho sakta hai.

True — iska sign sirf pivot ke relative entry ke sign ko reflect karta hai jo clear ho rahi hai; negative multiplier ka matlab hai tum effectively ek scaled pivot row add kar rahe ho.

Gaussian elimination ke liye matrix ka square hona zaroori hai.

False — ye kisi bhi

m \times n

system pe kaam karta hai; squareness unique solution tab guarantee karta hai jab full set of pivots bhi exist kare.

Spot the error

"Maine

A

ko triangularise kiya, phir original

b

ko back substitution mein plug kiya."

Error —

b

ko augmented matrix pe har row operation ke saath saath chalana chahiye; untouched original RHS use karna galat system deta hai.

a_{ik}

clear karne ke liye maine

R_{i} \to R_{i} + \frac{a _{ik}}{a _{k k}} R_{k}

use kiya."

Error — sign galat hai; ye minus hona chahiye (

R_{i} \to R_{i} - m_{ik} R_{k}

) taaki term

m_{ik} a_{k k}

a_{ik}

ko cancel kare, double karne ki jagah.

"Pivot

0

tha, to maine phir bhi divide kiya aur infinity mili, matlab infinitely many solutions."

Error — zero pivot division forbid karta hai; tumhe neeche ki kisi row mein se nonzero entry wali row swap karni chahiye, ya column skip karna chahiye. Yahan infinity ek bug hai, conclusion nahi.

"Maine diagonal ke upar se pivot choose kiya taaki numbers nicer hon."

Error — pivot apne column mein diagonal pe ya neeche se choose hota hai; upar ki row use karna already bani staircase ko toot deta hai aur cleared zeros wapas la sakta hai.

U

milne ke baad, maine row 3 ko

- 1

se scale kiya lekin uski RHS entry scale karna bhool gaya."

Error — row scale karne ka matlab hai poori row scale karna, augmented column sameit; warna woh equation ab kuch galat kehti hai.

"Elimination ke dauran do rows identical ho gayi, isliye system inconsistent hai."

False — identical rows subtract karne ke baad all-zero row produce karti hain (

0 = 0

), jiska matlab redundancy aur free variable hai, inconsistency nahi.

"Maine partial pivoting kiya sabse chhoti nonzero entry wali row swap karke."

Error — partial pivoting sabse badi magnitude entry wali row swap karti hai; chhote pivots rounding errors amplify karte hain (dekho Pivoting and Numerical Stability).

Why questions

Hum pivot row ka multiple subtract kyu karte hain, offending entry simply delete kyu nahi karte?

Kyunki hum sirf teeno legal operations use kar sakte hain, jo solution set intact rakhti hain; entry blank karna silently equation change kar dega.

Multiplier exactly

a_{ik} / a_{k k}

hi kyun define hota hai aur isse zyada kuch yaad karne ki zaroorat nahi?

New entry

a_{ik} - m_{ik} a_{k k}

ko zero set karke aur solve karne se

m_{ik} = a_{ik} / a_{k k}

force hota hai; formula literally "ise zero banao" ki requirement hai.

Back substitution bottom-up kyun jaati hai, top-down kyun nahi?

Sirf upper-triangular system ki last equation mein ek single unknown hota hai; use solve karne se next-to-last free ho jaata hai, isliye har step upar ek naya variable exactly reveal karta hai.

Hum augmented matrix ki taklif kyun uthate hain,

A

aur

b

alag-alag track karne ki jagah?

Taaki ek row operation coefficients aur RHS ko saath aur consistently transform kare — inhe pack karna woh classic mistake rokta hai jisme RHS bhool jaate hain.

Zero pivot ke saath jab us column mein neeche koi nonzero entry nahi — algorithm kyu nahi rukta?

Iska matlab hai woh variable kisi remaining equation se pin nahi hua — ek free variable; tum simply next column pe move karte ho aur staircase build karte rehte ho.

R_{i} \to R_{i} - m R_{j}

reversible kyun hai?

Kyunki

R_{i} \to R_{i} + m R_{j}

apply karna ise exactly undo karta hai; reversibility hi guarantee karti hai ki koi solutions create ya lose nahi hue.

Nonzero pivots ki count sirf solve karne se zyada kyun matter karti hai?

Ye matrix ka rank equals karta hai aur, pivots ke product ke zariye, Determinant feed karta hai — elimination system ko measure karta hai, sirf solve nahi karta.

Partial pivoting accuracy kyun improve karta hai jab pivot exactly zero nahi bhi ho?

Tiny pivot se divide karna floating-point errors magnify karta hai; bada entry swap karna multipliers ko small rakhta hai aur arithmetic stable rehti hai.

Edge cases

2 \times 2

system ek single all-zero row aur ek real equation mein reduce ho jaata hai — kitne solutions?

Infinitely many: do unknowns pe ek genuine constraint ek free variable chodta hai, solutions ki poori line.

Square system pe elimination ke baad har diagonal pivot nonzero hai — ye kya guarantee karta hai?

Unique solution: full pivots ka matlab full rank, nonzero Determinant, aur back substitution har variable exactly determine karta hai.

Pehla pivot

a_{11}

0

hai lekin kisi lower row mein nonzero first entry hai — eliminate karne se pehle kya karna chahiye?

Woh lower row ko pivot position mein swap karo; tab hi pivot se division valid hai aur column 1 clear ho sakta hai.

Ek system mein unknowns se zyada equations hain (

m > n

) aur elimination

[0 \dots 0 ∣ 7]

deta hai — verdict?

Inconsistent — ek extra equation baaki se contradict karta hai; over-determined systems mein commonly exact solution nahi hota.

Ek system mein unknowns se kam equations hain (

m < n

) aur ye consistent hai — kya iska unique solution ho sakta hai?

Nahi — unknowns se kam pivots hone par kam se kam ek variable free hoga, jo infinitely many solutions force karta hai (exactly one kabhi nahi).

Elimination ke dauran ek pivot column ke neeche already sab zeros hain — kaam ko kya hota hai?

Kuch clear nahi karna: column done hai, isliye next column pe move karo; us pivot ko koi operations ki zaroorat nahi thi.

Connections

Row Echelon Form / Reduced Row Echelon Form — woh target shapes jinke around ye traps revolve karte hain.
Solving Linear Systems — Consistency — no/one/infinite verdicts formally.
Pivoting and Numerical Stability — swapping aur pivot size kyun matter karti hai.
LU Decomposition — jahan multipliers store hote hain.
Rank of a Matrix / Determinant — pivots secretly kya measure karte hain.