Cramer's rule — Hinglish notes

4.5.24 · Maths › Linear Algebra (Full)

WHAT is Cramer's Rule?

Zaroori condition: yeh sirf tab kaam karta hai jab $== det A \neq = 0 ==$ ho (matrix invertible ho, isliye ek unique solution exist karta hai).

WHY does it work? (Derivation from scratch)

Hum ise ek fact use karke derive karte hain: determinant apni columns mein multilinear aur alternating hota hai, matlab $det$ ek "signed volume" ki tarah behave karta hai jo:

jab aap ek column ko scale karo toh scale hota hai,
jab aap ek column ko sum mein split karo toh add hota hai,
zero ho jaata hai agar do columns equal hoon.

Step 1 — Ek clever matrix banao. Identity matrix $I$ lo aur uski $i$ -th column ko unknown vector $x$ se replace karo. Ise $X_{i}$ kaho. Jaise $n = 3$ , $i = 2$ ke liye: $X_{2} = 100 x_{1} x_{2} x_{3} 001 .$ Yeh step kyun? Yeh matrix upper/lower-mixed hai lekin $i$ -th row/column ke along expand karne par $det X_{i} = x_{i}$ easily milta hai.

Step 2 — $det X_{i}$ compute karo. Special structure par determinant expand karte hue, $i$ -th ko chhodkar har column ek standard basis vector hai, isliye $det X_{i} = x_{i} .$ Yeh step kyun? Identity columns factor $1$ contribute karte hain; sirf diagonal par $x_{i}$ entry survive karta hai.

Step 3 — $A$ se multiply karo. Product $A X_{i}$ consider karo. $A$ ko kisi matrix se multiply karna $A$ ko har column se multiply karta hai:

Ek standard-basis column $e_{k}$ $A$ ka column $k$ pick out karta hai, yaani $a_{k}$ .
$i$ -th column $x$ se $A x = b$ milta hai.

Toh $A X_{i}$ exactly $A$ hai jisme $i$ -th column ko $b$ se replace kar diya gaya hai — yahi $== A_{i} ==$ hai: $A X_{i} = A_{i} .$ Yeh step kyun? Yahi core hai: matrix multiplication apne aap column ko hamare liye swap kar deta hai.

Step 4 — Determinants lo aur multiplicativity use karo ( $det (M N) = det M det N$ ): $det A \cdot det X_{i} = det A_{i} ⟹ det A \cdot x_{i} = det A_{i} .$

Step 5 — Solve karo. Agar $det A \neq = 0$ : $x_{i} = \frac{det A _{i}}{det A}$

HOW to use it — worked examples

Forecast-then-Verify

Common mistakes

Recall Feynman: explain to a 12-year-old

Socho ek recipe hai: ek smoothie ( $b$ ) banane ke liye tum banana, apple aur milk mix karte ho ( $A$ ki columns). Cramer's rule tumhe exactly batata hai ki kitne bananas chahiye. Trick yeh hai: original ingredient-box ka "size" measure karo (woh hai $det A$ ), phir ek naya box banao jisme banana ingredient ki jagah poora smoothie swap karo aur woh box measure karo ( $det A_{1}$ ). Dono sizes ko divide karo — woh ratio exactly bananas ki quantity hai! Har ingredient ke liye yahi karo. Agar original box flat ho (size $0$ ), toh ingredients "ek saath stuck" hain aur koi single recipe nahi hai.

Flashcards

Cramer's rule kya compute karta hai, aur yeh kab valid hai?

x_{i} = det A_{i} / det A

, valid sirf tab jab

det A \neq = 0

ho (unique solution).

Cramer's rule mein

A_{i}

kya hai?

A

jisme

i

-th column ko vector

b

se replace kar diya gaya ho.

Woh key identity jo Cramer's rule prove karta hai

A X_{i} = A_{i}

aur

det X_{i} = x_{i}

, isliye

det A \cdot x_{i} = det A_{i}

det A \neq = 0

kyun zaroori hai?

Warna zero se divide hoga; system ka koi unique solution nahi hoga.

det

ki kaun si property

det X_{i} = x_{i}

deti hai?

Multilinearity + identity columns ka basis vectors hona.

b

double karne par solution ka kya hota hai?

Har

x_{i}

double ho jaata hai (

A_{i}

ki columns

b

mein linear hain).

Cramer's rule bade numerical systems ke liye kharab kyun hai?

Cost

(n + 1)

determinants ki tarah grow karti hai aur tiny

det A

se divide karna unstable hai; Gaussian elimination faster/stabler hai.

Avoid karne wali common error

b

se column ki jagah row replace karna.

Connections

Determinants — engine; Cramer "determinant as volume" hai.
Matrix Inverse — Cramer ⇔ $A^{- 1} = \frac{1}{d e t A} adj (A)$ jo $b$ par apply hota hai.
Cofactor Expansion — $det A_{i}$ aise compute hote hain.
Gaussian Elimination — bade systems ke liye practical alternative.
Multilinear and Alternating Maps — woh properties jo proof ko kaam karti hain.
Invertible Matrix Theorem — $det A \neq = 0 ⟺$ unique solution.