Foundations — Determinants — cofactor expansion along any row - column

4.5.21 · D1 · Maths › Linear Algebra (Full) › Determinants — cofactor expansion along any row - column

Parent note on cofactor expansion padhne se pehle, aapko har woh symbol aana chahiye jo woh use karta hai. Neeche, har ek idea pichle ke upar build hota hai — upar se neeche padho, kuch bhi skip mat karo.

1. Ek matrix, aur uska size $n$

Ise picture karo ek spreadsheet ki tarah: rows ko upar se neeche $1,2,\dots ,n$ number karo aur columns ko left se right $1,2,\dots ,n$. Figure s01 hamare example matrix par in row aur column indices ko label karta hai.

Topic ko iske zaroorat kyun hai: determinant sirf square matrices ke liye defined hai, aur general formulas $1,2,\dots ,n$ par run karte hain — isliye koi bhi sum samajhne se pehle aapko pata hona chahiye ki $n$ kya count karta hai.

2. Entry $a_{ij}$ — ek number ko address karna

Ise ek street address ki tarah padho: $a_{23}$ matlab "row 2 par jao, phir column 3". Upar wali matrix mein, $a_{23}=5$ aur $a_{31}=1$.

Topic ko iske zaroorat kyun hai: parent note ke har formula mein specific entries $a_{ij}$ multiply hoti hain; aapko grid mein se exact number instantly nikalna aana chahiye.

3. Symbol $\det$ — "determinant of" operation

Humne abhi tak yeh nahi bataya ki badi matrix ke liye $\det A$ kaise compute karein — yahi parent note ka poora point hai. Yahan hum sirf symbol ka matlab fix kar rahe hain taaki baad ke formulas jaise $\det A=\dots$ padhe jaayein "number $\det A$ equals …". Aane wale sections (§4, §5) chote cases ke liye actual recipes dete hain.

Topic ko iske zaroorat kyun hai: parent note seedha "$\det A=\sum \dots$" se shuru hota hai — aap woh line tab tak nahi padh sakte jab tak $\det$ ka matlab kuch na ho aapke liye.

4. Sigma notation $\sum$ — "ek list add karo"

Topic ko iske zaroorat kyun hai: expansion formula $\det A={\sum }_{j=1}^n{a}_{ij}{C}_{ij}$ iske siwa kuch nahi hai "ek line ke saath chalo aur pieces add karo". $\sum$ ke bina aap ise padh hi nahi sakte.

5. $1\times 1$ aur $2\times 2$ determinants — base cases

Parent note jo bhi karta hai woh recurse karta hai neeche jab tak baaki matrices tiny na ho jaayein. Isliye humein sabse chote cases nail karne hain jahan $\det$ directly defined hai, koi recipe nahi chahiye.

Topic ko iske zaroorat kyun hai: cofactor expansion ek $3\times 3$ ko $2\times 2$ pieces mein todta hai, aur ek $4\times 4$ ko $3\times 3$ se $2\times 2$ se $1\times 1$ mein. Yeh do tiny cases recursion ka floor hain — recipe yahan rukti hai kyunki yeh directly defined hain.

6. Row aur column delete karna → Minor $M_{ij}$

Topic ko iske zaroorat kyun hai: minor woh chota determinant hai jo expansion ke har step par produce hota hai. Yahi woh hai jis se recursion physically hoti hai — ek $3\times 3$ minor-hunt $2\times 2$ leftovers deta hai, jo $1\times 1$ leftovers dete hain.

7. Sign $(-1{)}^{i+j}$ — checkerboard

$-1$ ko power tak raise karne se yeh kyun hota hai? Kyunki $-1$ ko even baar multiply karne par $+1$ milta hai, aur odd baar par $-1$. Isliye $i+j$ ki parity (even-ya-odd-ness) hi sirf matter karti hai.

Topic ko iske zaroorat kyun hai: is sign ke bina chote determinants galat number mein add hote. Yeh sign ek gehra fact encode karta hai (permutation parity, Leibniz formula for determinants se) — lekin computing ke liye, yeh simply checkerboard hai.

8. Cofactor $C_{ij}$ — signed minor

Topic ko iske zaroorat kyun hai: poori expansion entries times cofactors se bani hai. Cofactor formula ka atom hai.

9. Symbols ko saath mein lagana — row AUR column expansion

Pehle ek particular row fix karo, uska number $i$ rakhte hain (koi bhi single value $1$ se $n$ tak jo aap choose karo). Phir $j$ ko us row ke across sweep karo. Ab parent formula ka har piece defined hai:

$\det A=\underset{\text{§4 add up}}{\underbrace{\sum _{j=1}^n}}\underset{\text{§2 entry}}{\underbrace{a_{ij}}}\underset{\text{§7 sign}}{\underbrace{(-1{)}^{i+j}}}\underset{\text{§6 minor}}{\underbrace{M_{ij}}}(\text{row }i\text{ fixed})$

Zor se padho: "Ek row $i$ chuno. Us row ke saath chalo; har entry ke liye, uske checkerboard sign aur uske minor se multiply karo; sab add karo."

10. Yeh foundations topic ko kaise feed karte hain

Left pe har box solid hona chahiye tabhi right pe "Cofactor expansion" box sense karta hai.

Equipment checklist

Apne dimaag mein jawab do, phir reveal karo:

Connections

• Leibniz formula for determinants — checkerboard sign ka gehra source (permutation parity).
• Properties of determinants (multilinearity, alternation) — kyun rows delete/add karna is tarah behave karta hai.
• Determinant via row reduction — faster cousin jab yeh basics pata hon.
• Adjugate matrix and inverse — cofactors se bana ek poora matrix.
• Cramer's rule — cofactors se equations solve karna.
• Invertibility and singular matrices — $\det A\ne 0⟺$ invertible.