Determinants — cofactor expansion along any row - column

4.5.21 · Maths › Linear Algebra (Full)

3. First principles se derivation (YEH kyon kaam karta hai)

Row-1 expansion kaise milti hai. $n!$ permutations ko group karo by which column mein row-1 ki entry jaati hai, use $j = σ (1)$ kaho.

$det A = \sum_{j = 1}^{n} a_{1 j} \sum_{σ \in S_{n} σ (1) = j} sgn (σ) \prod_{k = 2}^{n} a_{k, σ (k)} .$

Yeh step kyon? Humne sirf ek bada sum $n$ buckets mein split kiya — koi term nahi gayi, koi add nahi hui.

Ab inner sum dekho. Column $j$ aur row $1$ hata dene ke baad, baaki $σ$ jo rows $2.. n$ tak restricted hai, leftover columns ke saari permutations pe range karta hai — yahi minor $M_{1 j}$ ki definition hai, bas ek sign ke alawa.

Sign: entry $(1, j)$ ko leading position mein laane ke liye chosen column ko $j - 1$ columns se hop karna padta hai; row $1$ ke saath mila ke yeh $(- 1)^{1 + j}$ factor deta hai. Isliye: $\sum_{σ (1) = j} sgn (σ) \prod_{k \geq 2} a_{k, σ (k)} = (- 1)^{1 + j} M_{1 j} = C_{1 j} .$

Yeh step kyon? Parity bookkeeping exactly wahi hai jo $(- 1)^{i + j}$ package karta hai. Wapas substitute karo: $det A = j = 1 \sum n a_{1 j} C_{1 j} .$

Koi bhi aur row/column: Leibniz sum symmetric hai — har row aur column identically appear karta hai. Do rows swap karne se $sgn$ flip hota hai lekin tum kisi bhi row ke basis pe pehle group kar sakte ho. Kyunki $det A = det A^{T}$ (transpose permutation sum ko invariant rakhta hai), column expansion = transpose ka row expansion. Isliye har line same answer deta hai. ∎

Figure — Determinants — cofactor expansion along any row - column

4. 80/20 — jo cheez actually marks dilaati hai

5. Worked examples

6. Steel-manned mistakes

7. Flashcards

Minor

M_{ij}

kya hota hai?

Row

i

aur column

j

delete karne ke baad matrix ka

det

Cofactor

C_{ij}

kya hota hai?

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

— signed minor.

Row

i

ke saath cofactor expansion batao.

det A = \sum_{j} a_{ij} (- 1)^{i + j} M_{ij}

Kya tum kisi bhi row ya column ke saath expand kar sakte ho?

Haan — sab same value dete hain (Leibniz sum &

det A = det A^{T}

se follow hota hai).

Cofactor

C_{12}

ka sign?

(- 1)^{1 + 2} = - 1

2 \times 2

determinant formula?

a d - b c

Expand karne ke liye best line (80/20)?

Woh row/column jisme sabse zyada zeros hon.

Do rows swap karne ka

det

pe effect?

Determinant ko

- 1

se multiply karta hai.

k \times

ek row doosri mein add karne ka effect?

det

mein koi change nahi.

(- 1)^{i + j}

kyon aata hai?

Yeh us permutation ki parity (sign) hai jo entry

(i, j)

ko leading corner mein move karti hai.

det A = 0

ka geometrically kya matlab hai?

Map space ko collapse karta hai →

A

singular/non-invertible hai.

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho ek badi numbers ki grid hai. "Determinant" ek khaas number hai jo batata hai ki koi shape kitna bada ya chhota hoti hai jab tum ise is grid se stretch karte ho. Ise dhundhne ke liye, tum top row ke saath chalte ho. Har number ke liye, apna haath rakho uski row aur column pe, haath ke neeche jo chhota number bache use dhundho, use checkerboard jaisa $+$ ya $-$ sign do, aur sab add karo. Cool baat: tum kisi bhi row ya column ke saath chal sakte ho — yahan tak ki zeros se bhari hui bhi (jo sabse aasaan hai, kyunki zeros matlab kam kaam!) — aur har baar same answer milega.

Connections

Leibniz formula for determinants — woh parent definition jisse humne derive kiya.
Determinant via row reduction — faster computational cousin.
Properties of determinants (multilinearity, alternation)
Adjugate matrix and inverse — $A^{- 1} = \frac{1}{d e t A} adj (A)$ , cofactors se bana.
Cramer's rule — cofactors use karke $A x = b$ solve karta hai.
Invertibility and singular matrices — $det A \neq = 0 ⟺$ invertible.

Determinants — cofactor expansion along any row - column

1. Building blocks

2. Expansion formula