Properties of determinants — Hinglish notes

2.6.9 · Maths › Matrices & Determinants — Introduction

Determinant KYA hai (taaki hum jaanein hum kya prove kar rahe hain)

Yeh definition KYU? Yeh wahi ek formula hai jo (a) har row mein linear hai aur (b) jab do rows equal hon to zero hai — aur yahi do facts neeche ki har property ko force karte hain.

Har property kaise follow karti hai (derive karo, yaad mat karo)

$A$ ko rows $R_{1}, R_{2}, \dots, R_{n}$ ke saath likho. $det$ ko ek function $D (R_{1}, \dots, R_{n})$ samjho.

P1. Transpose ko kuch nahi badalta

$det (A^{⊤}) = det (A)$ Kyu: Permutation sum mein, $a_{iσ (i)}$ vs $a_{σ (i) i}$ — $σ^{- 1}$ se reindex karne par same set of terms milte hain same sign ke saath (kyunki $sgn (σ) = sgn (σ^{- 1})$ ). Consequence: har row property ek column property bhi hai.

P2. Do rows swap karne se sign flip hoti hai

$D (\dots, R_{i}, \dots, R_{j}, \dots) = - D (\dots, R_{j}, \dots, R_{i}, \dots)$ Kyu: Ek swap har permutation ko ek transposition ke saath compose karta hai, har term ka $sgn$ flip karta hai.

P3. Do equal rows ⟹ determinant $= 0$

Kyu: Do equal rows swap karo: value apne $-$ (khud) ke barabar honi chahiye. To $det = - det \Rightarrow det = 0$ .

P4. Ek row ko $k$ se multiply karne par determinant $k$ se multiply hota hai

$D (\dots, k R_{i}, \dots) = k D (\dots, R_{i}, \dots)$ Kyu: Har term mein row $i$ se exactly ek factor hota hai, isliye $k$ unne sabse bahar aa jaata hai. Corollary: $det (k A) = k^{n} det A$ ( $n$ mein se har row ek $k$ contribute karti hai).

P5. Ek row mein additivity (linearity)

$D (\dots, R_{i} + R_{i}^{'}, \dots) = D (\dots, R_{i}, \dots) + D (\dots, R_{i}^{'}, \dots)$ Kyu: Har term row $i$ se jo ek entry leti hai usme linear hai.

P6. Ek row ka multiple doosri row mein add karne se value nahi badlti

$D (\dots, R_{i} + k R_{j}, \dots, R_{j}, \dots) = D (\dots, R_{i}, \dots, R_{j}, \dots)$ Kyu: P5 se yeh split hota hai original plus $k \cdot D (\dots, R_{j}, \dots, R_{j}, \dots)$ mein; doosre mein do equal rows hain, isliye woh $0$ hai (P3). Yeh haath se calculation ke liye sabse kaam aane wala rule hai.

P7. Zeros ki ek row ⟹ determinant $= 0$

Kyu: P4 mein $k = 0$ rakh do.

P8. Triangular matrix ⟹ diagonal ka product

$det = a_{11} a_{22} \dots a_{nn}$ Kyu: Ek triangular matrix mein, wahi ek permutation jo har row se nonzero entry chunti hai woh identity hai (koi bhi off-diagonal choice ek zero factor force karti hai).

P9. Product rule

$det (A B) = det (A) det (B)$ Kyu (sketch): Dono sides $A$ ki rows mein multilinear-alternating hain aur $I$ par normalized hain; axioms ek aisi unique function pin down karte hain, jo equality force karti hai. Consequence: $det (A^{- 1}) = \frac{1}{det A}$ .

Worked Examples

Recall Feynman: ek 12-saal ke bachche ko samjhao

Ek stretchy graph paper ki sheet imagine karo. Ek matrix use tedha aur stretch karta hai. Determinant woh number hai jo kehta hai "ek chhota sa square kitne times bada ho gaya?" Agar tum sheet palat do (mirror image), number negative ho jaata hai — yahi row-swap ka sign flip hai. Agar do directions ek hi taraf point karti hain, square zero area mein squash ho jaata hai — yahi "do equal rows give 0" hai. Ek row ka multiple doosri row mein add karna ek deck of cards ke upar ko sideways slide karne jaisa hai: shape jhuk jaata hai lekin area same rehta hai — isliye determinant nahi badlta.

Flashcards

Determinant kaunsi geometric quantity measure karta hai?

Linear map ke under area/volume ka (signed) scaling factor; sign orientation encode karta hai.

Do rows swap karne se determinant negative kyu hota hai?

Ek swap ek transposition hai, jo har permutation term ka sign flip karta hai. (P2)

Agar do rows equal hain, to det

= 0

kyu hai?

Equal rows swap karne par

det = - det

milta hai, isliye

det = 0

. (P3)

EK row ko

k

se multiply karne ka effect?

Determinant

k

se multiply hota hai. (P4)

Poore

n \times n

matrix ko

k

se multiply karne ka effect?

det (k A) = k^{n} det A

Kaun sa row operation determinant unchanged rehne deta hai?

Ek row ka multiple doosri row mein add karna:

R_{i} \to R_{i} + k R_{j}

. (P6)

det (A^{⊤}) = det A

kyu hota hai?

σ^{- 1}

se permutation sum ko reindex karne par identical signed terms milte hain. (P1)

Triangular matrix ka determinant?

Diagonal entries ka product. (P8)

Product rule aur inverses ke liye uska corollary batao.

det (A B) = det A det B

; isliye

det (A^{- 1}) = 1/ det A

Kya

det (A + B) = det A + det B

sach hai?

Nahi — linearity sirf per-row hold karta hai jab baaki rows fixed hon.

Haath se numeric determinant compute karne ka sabse tez tarika?

P6 use karke ek triangular matrix banao, phir diagonal multiply karo (P8).

Connections

Determinant — Definition and Expansion by Minors
Cofactors and Adjoint of a Matrix
Inverse of a Matrix via Adjoint
Cramer's Rule and Systems of Linear Equations
Elementary Row Operations & Rank
Area of a Triangle using Determinants
Linear Transformations & Scaling of Area