Determinant of 3×3 matrix — cofactor expansion

2.6.8 · Maths › Matrices & Determinants — Introduction

Isse is tarah socho: tum 3D space mein ek parallelepiped ka "signed volume" measure kar rahe ho. Ek baar mein sab compute karne ke bajaye, tum usse ek direction ke along slice karte ho aur har slice ke contributions ko add up karte ho.

Cofactor expansion kya hai?

Element $a_{ij}$ ka cofactor $C_{ij}$ hota hai: $C_{ij}=(-1{)}^{i+j}{M}_{ij}$ jahan $M_{ij}$ minor hai (2×2 matrix ka determinant jo row $i$ aur column $j$ delete karne ke baad milta hai).

Determinant hota hai: $\det (A)={\sum }_{j=1}^3{a}_{ij}{C}_{ij}\text{(expanding along row }i\text{)}$ ya $\det (A)={\sum }_{i=1}^3{a}_{ij}{C}_{ij}\text{(expanding along column }j\text{)}$

YEH kaam kyun karta hai?

Determinant measure karta hai ki ek linear transformation volumes ko kitna scale karta hai. Jab hum kisi row ke along expand karte hain, to hum transformation ko simpler transformations (2×2 minors) ki linear combination ke roop mein express kar rahe hote hain, us row ke elements se weighted hokar. Alternating signs $(-1{)}^{i+j}$ correct orientation (positive/negative volume) ensure karte hain.

First principles se derivation:

Determinant ki permutation definition se shuru karo: $\det (A)={\sum }_{\sigma \in S_3}\text{sgn}(\sigma )a_{1\sigma (1)}{a}_{2\sigma (2)}{a}_{3\sigma (3)}$

Terms ko us first row element ke hisaab se group karo jise woh contain karte hain:

• $a_{11}$ wale terms: saari permutations jo column 1 se shuru hoti hain
• $a_{12}$ wale terms: saari permutations jo column 2 se shuru hoti hain
• $a_{13}$ wale terms: saari permutations jo column 3 se shuru hoti hain

$a_{11}$ wale terms ke liye, usse factor out karne ke baad, bacha hua product sirf rows 2, 3 aur columns 2, 3 ke elements involve karta hai—bilkul minor $M_{11}$ ki tarah. Permutation se aane wala sign $(-1{)}^{1+1}=+1$ deta hai.

Isi tarah $a_{12}$ aur $a_{13}$ ke liye, result milta hai: $\det (A)=a_{11}(-1{)}^{1+1}{M}_{11}+a_{12}(-1{)}^{1+2}{M}_{12}+a_{13}(-1{)}^{1+3}{M}_{13}$

Yahi row 1 ke along cofactor expansion hai.

Sign pattern (checkerboard): $\left[\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right]$

Position $(i,j)$ ke liye sign hai $(-1{)}^{i+j}$.

KAISE compute karein: Step-by-step

Step 1: Ek row ya column choose karo (simplify karne ke liye zeros wala chunna).

Step 2: Us row/column ke har element ke liye:

• $(-1{)}^{i+j}$ se sign determine karo (ya checkerboard pattern use karo)
• 2×2 minor paane ke liye us element ki row aur column cross out karo
• 2×2 determinant compute karo
• Multiply karo: element × sign × minor

Step 3: Teeno contributions ka sum karo.

Row 1 kyun? Hum koi bhi row/column choose kar sakte hain—pehle standard approach demonstrate karte hain.

Row 1 ke along expand karte hue: $\det (A)=2\cdot C_{11}+(-3)\cdot C_{12}+1\cdot C_{13}$

$C_{11}$ compute karo:

• Sign: $(-1{)}^{1+1}=+1$
• Minor $M_{11}$: row 1, column 1 delete karo: $M_{11}=\left|\begin{array}{cc}0& 6\\ 4& 2\end{array}\right|=0\cdot 2-6\cdot 4=-24$
• $C_{11}=(+1)(-24)=-24$

Yeh step kyun? Hum overall volume scaling mein element $a_{11}=2$ ke contribution ko isolate kar rahe hain. Minor capture karta hai ki baaki ki 2D "slice" kaise behave karti hai.

$C_{12}$ compute karo:

• Sign: $(-1{)}^{1+2}=-1$
• Minor $M_{12}$: row 1, column 2 delete karo: $M_{12}=\left|\begin{array}{cc}5& 6\\ -1& 2\end{array}\right|=5\cdot 2-6\cdot (-1)=10+6=16$
• $C_{12}=(-1)(16)=-16$

$C_{13}$ compute karo:

• Sign: $(-1{)}^{1+3}=+1$
• Minor $M_{13}$: row 1, column 3 delete karo: $M_{13}=\left|\begin{array}{cc}5& 0\\ -1& 4\end{array}\right|=5\cdot 4-0\cdot (-1)=20$
• $C_{13}=(+1)(20)=20$

Final sum: $\det (A)=2(-24)+(-3)(-16)+1(20)=-48+48+20=20$

Column 2 kyun? Dekho column 2 mein DO zeros hain! Iska matlab sirf EK minor compute karna padega teen ki jagah.

Column 2 ke along expand karte hue: $\det (B)=0\cdot C_{12}+5\cdot C_{22}+0\cdot C_{32}=5\cdot C_{22}$

$C_{22}$ compute karo:

• Sign: $(-1{)}^{2+2}=+1$
• Minor $M_{22}$: row 2, column 2 delete karo: $M_{22}=\left|\begin{array}{cc}3& 2\\ 4& 7\end{array}\right|=3\cdot 7-2\cdot 4=21-8=13$
• $C_{22}=(+1)(13)=13$

Final: $\det (B)=5\cdot 13=65$

Yeh strategy kyun? Zeros poore terms eliminate kar dete hain, computation drastically reduce ho jaata hai. Hamesha scan karo ki kis row/column mein sabse zyada zeros hain.

Column 3 ke along expand karte hue: $\det (C)=3\cdot C_{13}+6\cdot C_{23}+9\cdot C_{33}$

$C_{13}$ compute karo:

• Sign: $(-1{)}^{1+3}=+1$
• $M_{13}=\left|\begin{array}{cc}4& 5\\ 7& 8\end{array}\right|=32-35=-3$
• $C_{13}=+1\cdot (-3)=-3$

$C_{23}$ compute karo:

• Sign: $(-1{)}^{2+3}=-1$
• $M_{23}=\left|\begin{array}{cc}1& 2\\ 7& 8\end{array}\right|=8-14=-6$
• $C_{23}=-1\cdot (-6)=6$

$C_{33}$ compute karo:

• Sign: $(-1{)}^{3+3}=+1$
• $M_{33}=\left|\begin{array}{cc}1& 2\\ 4& 5\end{array}\right|=5-8=-3$
• $C_{33}=+1\cdot (-3)=-3$

Final: $\det (C)=3(-3)+6(6)+9(-3)=-9+36-27=0$

Zero kyun? Is matrix ki linearly dependent rows hain: dekho ki $\text{row 3}=2\cdot \text{row 2}-\text{row 1}$, kyunki $2(4,5,6)-(1,2,3)=(7,8,9)$. Kyunki teeno rows linearly dependent hain, woh sirf ek 2D plane (flat sheet) mein hain, isliye jo parallelepiped woh span karti hain uska volume zero hai. Zero determinant matlab transformation 3D space ko 2D mein collapse kar deta hai.

Common mistakes

Yeh sahi kyun lagta hai: Signs ke bina formula zyada simple lagta hai.

Fix: Signs $(-1{)}^{i+j}$ optional nahi hain—yeh encode karte hain ki har contribution total signed volume mein add hoga ya subtract. Yaad rakhne ke liye checkerboard pattern use karo.

Yeh sahi kyun lagta hai: Indices confuse ho jaati hain, especially jab tezi se kaam kar rahe ho.

Fix: Double-check karo: "Main $a_{ij}$ ka cofactor dhundh raha hoon, isliye row $i$ AUR column $j$ cross out karunga."

Sach: Sabhi expansions same determinant dete hain—yeh matrix ki ek unique property hai. Alag rows/columns choose karna sirf ek computational strategy hai.

Yeh galat kyun lagta hai: Alag rows alag intermediate calculations produce karti hain, jisse lagta hai result bhi alag hona chahiye.

Fix: Determinant invariant hai. Alag paths, same destination.

Sahi: $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$

Fix: Yaad rakho "down-right minus up-left" ya "main diagonal minus anti-diagonal."

Practice problems

1. $A=\left[\begin{array}{ccc}1& 3& 2\\ -1& 0& 4\\ 2& 5& 1\end{array}\right]$ ke liye row 2 ke along expand karke $\det (A)$ compute karo.

2. $B=\left[\begin{array}{ccc}2& 0& 0\\ 3& 1& 0\\ 5& 4& 6\end{array}\right]$ (lower-triangular matrix) ke liye $\det (B)$ nikalo (row 1 ke along expand karo aur pattern notice karo: determinant diagonal entries ke product ke barabar hota hai $2\cdot 1\cdot 6=12$).

3. Dikhao ki $A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 4& 5\\ 0& 0& 6\end{array}\right]$ ko column 1 ke along expand karne par wahi result milta hai jo row 1 ke along expand karne par milta hai.

Ek bade 3×3 matrix ke liye, ek hi baar mein yeh compute karne ke bajaye, hum ek clever trick use karte hain: hum ek row choose karte hain, aur us row ke har number ke liye:

1. Apne haathon se us number ki row aur column dhak do
2. Bacha hua 2×2 grid dekho aur uska determinant compute karo (jo tumhe pehle se aata hai!)
3. Us 2×2 determinant ko original number se multiply karo
4. Ek checkerboard pattern ke hisaab se + ya − sign lagao

Phir teeno pieces add kar do. Jaadu yeh hai ki chahe tum kaunsi bhi row chuno, same answer milega! Yeh ek pahaad chadne jaisa hai: alag raaste, same summit.

Signs ka checkerboard pattern box ki "handedness" jaisa hai—kabhi kabhi transformation box ko andar-bahar palat deta hai, aur signs us cheez ko track karte hain.

Connections

• 2.6.01-What-is-a-determinant — foundational definition aur geometric meaning
• 2.6.05-Determinant-of-2×2-matrix — cofactor expansion ka base case
• 2.6.09-Properties-of-determinants — shortcuts jo cofactor expansion ko faster banate hain
• 2.6.11-Determinant-and-matrix-inverse — $\det (A)\ne 0$ iff $A$ invertible hai
• 2.7.03-Cramers-rule — systems solve karne ke liye cofactor expansion use karta hai
• 3.4.02-Cross-product-and-determinants — 3D cross product ek 3×3 determinant ke roop mein

#flashcards/maths

Matrix mein element $a_{ij}$ ka cofactor $C_{ij}$ kya hota hai? :: $C_{ij}=(-1{)}^{i+j}{M}_{ij}$ jahan $M_{ij}$ minor hai (row $i$ aur column $j$ delete karne ke baad bani matrix ka determinant)

3×3 matrix mein cofactors ka sign pattern (checkerboard) kya hota hai? :: $\left[\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right]$, $(-1{)}^{i+j}$ follow karte hue

3×3 matrix ke liye $M_{23}$ ka matlab kya hai? :: Minor $M_{23}$ wo 2×2 matrix ka determinant hai jo row 2 aur column 3 delete karne se milta hai

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