2.6.9 · D3Properties of determinants
Intuition Yeh page kis kaam ki haiProperties of determinants ne tumhe aath rules diye. Yeh page har tarah ka determinant problem jो ho sakta hai tumhare saamne rakhti hai aur dikhati hai ki kaun sa rule use sabse jaldi solve karta hai. Har solution se pehle tum forecast karoge — pehle guess karna hi woh tarika hai jisse rules yaad rehte hain.
Har determinant problem jo tum dekhte ho, in "cells" mein se ek hota hai. Hum sab ko cover karenge.
Cell
Kya trigger karta hai
Kaun sa rule jeet ta hai
Example
A — Pehle zeros banao, phir diagonal padhodense numbers, koi obvious pattern nahi
P6 + P8
Ex 1
B — Instant zero spot karodo equal / proportional rows ya cols
P3 (+P4)
Ex 2
C — Zero row / degenerateek poori row ya column ke zeros
P7
Ex 3
D — Common factors bahar nikaloek row/col ek clean multiple hai
P4
Ex 4
E — Swaps ke under sign track karosimplify karne ke liye rows rearrange karte ho
P2
Ex 5
F — Whole-matrix scaling (trap)det ( k A ) det ( A + B ) P4 corollary, P5 caveat
Ex 6
G — Product & inversedet ( A B ) det ( A − 1 ) P9
Ex 7
H — Real-world word problemarea / volume scaling
geometry of det
Ex 8
I — Exam twist (symbolic/letters)entries a , b , c x
P6 factoring
Ex 9
Har cell ko aise samjho: "kaun sa ek fact hai jo saari arithmetic ko gaa yab kar deta hai?"
Δ = 2 4 6 1 3 5 3 8 16
Forecast: padhne se pehle guess karo. Kya yeh ek bada number hai, chhota number hai, ya zero? Rows lagta hai jaise har agli row roughly double hai... dhyan rakho.
Step 1. R 2 → R 2 − 2 R 1 R 3 → R 3 − 3 R 1 Yeh step kyun? P6 (ek row ka multiple doosri row mein add karna) value nahi badalta , aur column 1 mein zeros bana deta hai — ek mushkil 3 × 3 = 2 0 0 1 1 2 3 2 7
Step 2. R 3 → R 3 − 2 R 2 Yeh step kyun? Same rule, ab diagonal ke neeche ki last entry clear kar rahe hain.
= 2 0 0 1 1 0 3 2 3
Step 3. Upper-triangular ⟹ diagonal multiply karo (P8).
Δ = 2 ⋅ 1 ⋅ 3 = 6.
Verify: original ke column 1 ke saath direct expansion: 2 ( 3 ⋅ 16 − 8 ⋅ 5 ) − 1 ( 4 ⋅ 16 − 8 ⋅ 6 ) + 3 ( 4 ⋅ 5 − 3 ⋅ 6 ) = 2 ( 48 − 40 ) − 1 ( 64 − 48 ) + 3 ( 20 − 18 ) = 16 − 16 + 6 = 6.
3 1 7 6 2 14 5 9 4
Forecast: column 2 ko column 1 ke saath ghoor ke dekho. Kuch double dikhta hai?
Step 1. Column 2 = ( 6 , 2 , 14 ) = 2 ⋅ ( 3 , 1 , 7 ) = 2 ⋅ Yeh step kyun? Ek column doosre column ka scalar multiple hai. P4 use karke column 2 se 2 = 2 3 1 7 3 1 7 5 9 4
Step 2. Ab columns 1 aur 2 equal hain.
Yeh step kyun? Equal columns determinant ko 0 = 2 ⋅ 0 = 0.
Verify: proportional columns hamesha 0 0
5 0 1 − 2 0 8 7 0 3
Forecast: ek poori row zeros hai. Answer kya hona chahiye, aur geometrically kyun?
Step 1. Row 2 mein sab zeros hain.
Yeh step kyun? P7: zero row det = 0 k = 0 0 0 = 0.
Verify: row 2 ke saath expand karo — har cofactor ek 0 0 0
3 1 2 6 0 2 9 2 0
Forecast: row 1 hai 3 × ( 1 , 2 , 3 ) 3 3
Step 1. Row 1 = 3 ( 1 , 2 , 3 ) 3 Yeh step kyun? Chhote numbers = kam arithmetic mistakes. P4 ek common row factor ko front mein laane deta hai.
= 3 1 1 2 2 0 2 3 2 0
Step 2. R 2 → R 2 − R 1 R 3 → R 3 − 2 R 1 Yeh step kyun? Aasaan expansion ke liye top ke neeche column 1 clear karo.
= 3 1 0 0 2 − 2 − 2 3 − 1 − 6
Step 3. Column 1 ke saath expand karo: sirf top-left survive karta hai.
Yeh step kyun? Column 1 mein do zeros expansion ko ek term tak reduce kar dete hain.
= 3 ⋅ 1 ⋅ [ ( − 2 ) ( − 6 ) − ( − 1 ) ( − 2 ) ] = 3 ( 12 − 2 ) = 3 ⋅ 10 = 30.
Verify: original ka row 1 ke saath direct expansion: 3 ( 0 ⋅ 0 − 2 ⋅ 2 ) − 6 ( 1 ⋅ 0 − 2 ⋅ 2 ) + 9 ( 1 ⋅ 2 − 0 ⋅ 2 ) = 3 ( − 4 ) − 6 ( − 4 ) + 9 ( 2 ) = − 12 + 24 + 18 = 30.
0 0 2 0 5 7 4 1 3
Forecast: yeh almost triangular hai lekin galat taraf (zeros bottom-left ki jagah top-left mein hain). Agar hum ise triangular form mein flip karein, sign ka kya hoga?
Step 1. R 1 ↔ R 3 Yeh step kyun? Isse zeros lower-left mein aa jaate hain, upper-triangular form milti hai — lekin har swap answer ko − 1 (P2). Track karo: running factor = − 1 = − 2 0 0 7 5 0 3 1 4
Step 2. Ab upper-triangular ⟹ diagonal ka product (P8).
Yeh step kyun? Diagonal read-off, koi expansion nahi.
= − ( 2 ⋅ 5 ⋅ 4 ) = − 40.
Verify: original lower -anti form mein hai; column 1 ke saath expand karo (sirf row 3 mein 2 ( − 1 ) 3 + 1 = + 2 ⋅ 0 5 4 1 = 2 ( 0 − 20 ) = − 40. + 40 − 40
Maano A = ( 1 3 2 4 ) det A = 1 ⋅ 4 − 2 ⋅ 3 = − 2 det ( 3 A ) det ( A + A ) = det A + det A
Forecast: bahut se students kehte hain det ( 3 A ) = 3 det A = − 6
Step 1 (a). 3 A dono rows ko 3 Yeh step kyun? P4 har row ke liye ek factor k n = 2 det ( k A ) = k n det A det ( 3 A ) = 3 2 det A = 9 ⋅ ( − 2 ) = − 18.
Step 2 (b). A + A = 2 A det ( 2 A ) = 2 2 ( − 2 ) = − 8 Yeh step kyun? Left side honest tarike se compute karo.
Step 3 (b). Right side: det A + det A = ( − 2 ) + ( − 2 ) = − 4 Yeh step kyun? Compare karo. Kyunki − 8 = − 4 galat hai. P5 ek row at a time mein linearity hai, poori matrix mein nahi.
Verify: 3 A = ( 3 9 6 12 ) det = 3 ⋅ 12 − 6 ⋅ 9 = 36 − 54 = − 18. 2 A = ( 2 6 4 8 ) det = 16 − 24 = − 8 = − 4.
A = ( 2 1 0 3 ) B = ( 1 0 4 2 ) det ( A B ) det ( A − 1 )
Forecast: det A det B det ( A B )
Step 1. det A = 2 ⋅ 3 = 6 det B = 1 ⋅ 2 = 2 Yeh step kyun? Diagonals padho; koi kaam nahi.
Step 2. Product rule P9: det ( A B ) = det A ⋅ det B = 6 ⋅ 2 = 12. Yeh step kyun? P9 kehta hai ki scaling factors multiply hote hain — 6 2 12
Step 3. Inverse: det ( A − 1 ) = det A 1 = 6 1 . Yeh step kyun? P9 se A − 1 A = I det I = 1 det ( A − 1 ) det A = 1
Verify: A B = ( 2 1 0 3 ) ( 1 0 4 2 ) = ( 2 1 8 10 ) det = 2 ⋅ 10 − 8 ⋅ 1 = 20 − 8 = 12. 6 1 ⋅ 6 = 1.
Ek designer graph paper par ek triangle draw karti hai jiske corners P ( 1 , 1 ) Q ( 5 , 2 ) R ( 2 , 6 ) M = ( 2 0 0 3 ) × 2 × 3 Stretched triangle ka area kya hai?
Forecast: machine width 2 3
Step 1. Determinant formula se original area (dekho Area of a Triangle using Determinants ):
Area = 2 1 Q x − P x R x − P x Q y − P y R y − P y = 2 1 4 1 1 5 . Yeh step kyun? P
Step 2. Evaluate: 4 1 1 5 = 20 − 1 = 19 = 2 19 = 9.5 Yeh step kyun? Seedha 2 × 2
Step 3. Machine apply karo. Linear Transformations & Scaling of Area se, area ∣ det M ∣ det M = 2 ⋅ 3 = 6. Yeh step kyun? P8 (triangular) det M ∣ det M ∣ hi area multiplier hai — determinant ka poora point yahi hai.
Step 4. Stretched area = 6 × 9.5 = 57. Yeh step kyun? Original area ko scaling factor se multiply karo.
Verify (units & sanity): areas square units mein hain. Width-2 3 2 × 3 = 6 det M 9.5 × 6 = 57 6 ×
Brute expansion ke bina prove karo:
1 1 1 a b c a 2 b 2 c 2 = ( b − a ) ( c − a ) ( c − b ) . Vandermonde determinant hai.)
Forecast: agar a = b zaroor ek factor ( b − a )
Step 1. R 2 → R 2 − R 1 R 3 → R 3 − R 1 Yeh step kyun? Pehli row subtract karna column 1 mein zero create karta hai aur common factors reveal karta hai.
= 1 0 0 a b − a c − a a 2 b 2 − a 2 c 2 − a 2 .
Step 2. Row 2 se ( b − a ) ( c − a ) x 2 − a 2 = ( x − a ) ( x + a ) Yeh step kyun? P4 ek poori row se common factor bahar kheench ta hai.
= ( b − a ) ( c − a ) 1 0 0 a 1 1 a 2 b + a c + a .
Step 3. Column 1 ke saath expand karo (sirf top-left 1 2 × 2 1 1 b + a c + a = ( c + a ) − ( b + a ) = c − b . Yeh step kyun? Column 1 mein do zeros expansion ko ek 2 × 2
Step 4. Combine karo.
= ( b − a ) ( c − a ) ( c − b ) .
Verify: a = 1 , b = 2 , c = 4 ( 2 − 1 ) ( 4 − 1 ) ( 4 − 2 ) = 1 ⋅ 3 ⋅ 2 = 6 1 1 1 1 2 4 1 4 16 = 1 ( 2 ⋅ 16 − 4 ⋅ 4 ) − 1 ( 1 ⋅ 16 − 4 ⋅ 1 ) + 1 ( 1 ⋅ 4 − 2 ⋅ 1 ) = 1 ( 16 ) − 1 ( 12 ) + 1 ( 2 ) = 6.
Common mistake Teen traps jinke against yeh page tumhe protect karti hai
F-trap: det ( k A ) = k n det A k det A Har row ko scale karna k har baar scale karta hai.E-trap: swap par sign flip bhool jaana (Ex 5). Ek swap = × ( − 1 ) F-trap 2: det ( A + B ) = det A + det B
Recall Kaun sa rule kaun si cell ko solve karta hai?Dense numbers, koi pattern nahi ::: P6 se triangularise karo, phir P8 (Ex 1, Cell A)
Do equal ya proportional rows/columns ::: P4 phir P3 → 0 0 det ( k A ) n × n k n det A det ( A B ) det ( A − 1 ) det A det B 1/ det A M ∣ det M ∣ R 1
Mera problem kaun si cell mein hai?
Cell B: P4 phir P3 gives zero
Cell F: k to the n, no additivity
Cell H: multiply area by det
Cell I: subtract rows and factor