Determinant of 3×3 matrix — cofactor expansion: Foundations

2.6.8 · D1 · Maths › Matrices & Determinants — Introduction › Determinant of 3×3 matrix — cofactor expansion

Is page par koi bhi assumption nahi hai. Parent note 2.6.8 padhne se pehle, tumhe kuch symbols aur pictures mein fluent hona zaroori hai. Hum har ek cheez ek-ek karke, unke dependency order mein, ground up se build karenge.

1. Matrix kya hai, aur chhote numbers ka matlab kya hai

$A = 25 - 1 - 3 04 162$

Isme 3 rows aur 3 columns hain, isliye hum ise 3×3 matrix kehte hain (kaho "three by three": pehle rows, phir columns).

Picture: ek spreadsheet socho, ya ek chhoti cinema mein seats — 3 rows deep, 3 seats wide.

Ab, hum ek specific number ko kaise point karte hain? Hum ek subscript use karte hain — woh chhote numbers jo ek letter ke neeche aur daayein taraf likhe jaate hain.

Figure dekho. Entry $a_{23}$ row 2, column 3 mein hai — woh upar matrix $A$ mein $6$ hai. Entry $a_{31}$ row 3, column 1 hai — woh $- 1$ hai.

Yeh topic isko kyun chahta hai: cofactor expansion baar baar "row $i$ aur column $j$ delete karo" aur "sign $(- 1)^{i + j}$ " ki baat karta hai. Agar tum instantly $a_{ij}$ locate nahi kar sakte, toh kuch bhi samajh nahi aayega.

2. Do vertical bars — determinant notation

Tum wohi numbers ka grid do alag tareekon se likhte dekkhoge:

$[25 - 3 0] versus 25 - 3 0$

Hum determinant ko $det (A)$ bhi likhte hain. Toh yeh teeno same single number ko represent karte hain:

$det (A) = ∣ A ∣ = 25 - 1 - 3 04 162$

Picture: brackets = ingredients ka box; bars = ek number jo tum unse banate ho.

Yeh topic isko kyun chahta hai: har minor $M_{ij}$ vertical bars ke saath likha jaata hai — woh khud ek determinant hai, matrix nahi. Dono ko confuse karna poori formula ko nonsense bana deta hai. "Yeh number ka matlab kya hai" ki deeper story ke liye 2.6.01-What-is-a-determinant dekho.

3. 2×2 determinant — har cheez ka atom

Parent note mein sab kuch aakhirkar 2×2 determinants compute karne par aa jaata hai, isliye hamen yeh perfectly yaad hona chahiye.

Yeh words mein kya kehta hai: main diagonal (top-left se bottom-right: $a$ aur $d$ ) ke do numbers ko multiply karo, phir anti-diagonal (top-right se bottom-left: $b$ aur $c$ ) ke product ko subtract karo.

" $a d - b c$ " kyun aur " $a d + b c$ " kyun nahi? 2×2 ka determinant do rows se bane parallelogram ka signed area hai (jo arrows sochke origin se). Subtraction wahi hai jo area ko negative banata hai jab do arrows swap hote hain — yeh orientation record karta hai (kaun sa arrow "left par" hai). Amber parallelogram figure mein dekho: uska area exactly $∣ a d - b c ∣$ hai, aur sign batata hai ki doosra arrow pehle se clockwise hai ya counter-clockwise. Poori kahani 2.6.05-Determinant-of-2×2-matrix mein hai.

Yeh topic isko kyun chahta hai: cofactor expansion ek 3×3 determinant ko teen aise 2×2 determinants mein convert karta hai. Yahi woh tool hai jis par poora method bana hai.

4. Row aur column delete karna — minor $M_{ij}$

Yeh cofactor expansion ki central move hai, aur yeh purely visual hai.

Figure follow karo. $M_{12}$ (row 1, column 2 ki position ka minor) nikalne ke liye:

Row 1 par ungli rakho — poori row cross out karo.
Column 2 par ungli rakho — poori column cross out karo.
Chaar numbers bachte hain, corners mein. Unhe apni original positions mein ek 2×2 matrix ke roop mein padho.
Iska determinant lo.

Hamare matrix $A$ ke liye: $M_{12} = 5 - 1 62 = (5) (2) - (6) (- 1) = 10 + 6 = 16$

Yeh topic isko kyun chahta hai: minors woh "chhote pieces" hain jo hum pehle se compute karna jaante hain. Expansion ka har term (ek number) × (ek sign) × (ek minor) hota hai.

5. Sign $(- 1)^{i + j}$ — checkerboard

Aakhri symbol $- 1$ ki ek chhoti si power hai.

Kyun? $- 1$ ko khud se even number of times multiply karne par $+ 1$ milta hai; odd number of times karne par ek $- 1$ bachta hai. Example: $(- 1)^{1 + 2} = (- 1)^{3} = - 1$ ; aur $(- 1)^{2 + 2} = (- 1)^{4} = + 1$ .

Practice mein tumhe power compute karne ki zaroorat nahi hai. Bas checkerboard yaad karo, jahan top-left hamesha $+$ hota hai:

$+ - + - + - + - +$

Picture: ek chessboard. Corner square white (+) hai, aur colour har baar flip hota hai jab tum ek square sideways ya up/down step karte ho.

Toh cofactor bas apna sahi plus ya minus pehne hua minor hai.

Yeh topic isko kyun chahta hai: signs decoration nahi hain — yeh orientation record karte hain (positive vs negative volume), bilkul jaise area ke liye $a d - b c$ mein minus ne kiya. Inhe drop karo aur tum har baar galat number paoge.

6. Greek letters aur sum symbol

Parent note ki derivation mein "scary-looking" shorthand ke teen pieces hain. Yeh sab hai, demystified.

Yeh bas "har column ke liye yeh karo aur total karo" likhne ka compact tarika hai.

Parent note ka proof section $σ$ (ek permutation — columns $1, 2, 3$ ki reshuffling) aur $sgn (σ)$ (iska sign, $+ 1$ ya $- 1$ ) bhi mention karta hai. Cofactor expansion use karne ke liye tumhe yeh nahi chahiye — yeh sirf "yeh kyun kaam karta hai" derivation mein aate hain. Agar yeh tumhe darrate hain, pehli baar padhte waqt us block ko skip karo; recipe apne aap khadi hai.

Number khud aur iska matlab: 2.6.01-What-is-a-determinant
2×2 atom poori tarah se: 2.6.05-Determinant-of-2×2-matrix
Zeros aur dependence ka faida uthane wale shortcuts: 2.6.09-Properties-of-determinants
Determinants se matrices invert karna: 2.6.11-Determinant-and-matrix-inverse
Unse equations solve karna: 2.7.03-Cramers-rule
3D geometry ka link: 3.4.02-Cross-product-and-determinants

Equipment checklist

Apne aap ko test karo — answer out loud bolo, phir reveal karo.

a_{ij}

mein, kaun sa subscript row hai?

Pehla wala,

i

. Doosra column hai,

j

a_{23}

kya point karta hai?

Row 2, column 3 mein entry.

[\dots]

aur

\dots

mein kya farq hai?

Square brackets = matrix (ek object); vertical bars = uska determinant (ek number).

2×2 determinant formula batao.

a c b d = a d - b c

(main diagonal minus anti-diagonal).

5 - 1 04

compute karo.

(5) (4) - (0) (- 1) = 20

Minor

M_{31}

nikalne ke liye tum kya cross out karte ho?

Row 3 aur column 1; bache hue 2×2 ka determinant lo.

(- 1)^{i + j}

kya hota hai jab

i + j

odd ho?

- 1

. Jab even ho,

+ 1

Checkerboard mein top-left sign kya hai?

Hamesha

+

Minor ke terms mein cofactor

C_{ij}

likho.

C_{ij} = (- 1)^{i + j} M_{ij}

\sum_{j = 1}^{3} a_{1 j} C_{1 j}

expand karke kya milta hai?

a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13}

3×3 determinant zero kyun ho sakta hai?

Rows linearly dependent hain — woh ek flat sheet mein lie karti hain, isliye spanned box ka zero volume hota hai.

Foundations — Determinant of 3×3 matrix — cofactor expansion

1. Matrix kya hai, aur chhote numbers ka matlab kya hai

2. Do vertical bars — determinant notation

3. 2×2 determinant — har cheez ka atom

4. Row aur column delete karna — minor $M_{ij}$

5. Sign $(- 1)^{i + j}$ — checkerboard

6. Greek letters aur sum symbol

7. Linear dependence — kyun kuch determinants zero hote hain

Prerequisite map

Yeh aage kahan jaate hain

Equipment checklist