Determinant of 2×2 matrix: Worked examples — Hinglish deep dive

2.6.7 · D3 · Maths › Matrices & Determinants — Introduction › Determinant of 2×2 matrix

Yeh page Determinant of 2×2 matrix ki practice arena hai. Kuch bhi compute karne se pehle, ek baar woh ek rule yaad kar lete hain jis par sab kuch tika hua hai.

Yahan har symbol ka ek seedha matlab hai:

$a, b, c, d$ bas woh chaar numbers hain jo box ke andar likhe hain.
$det (A)$ (bolo "det of A") woh ek akela number hai jo tumhe milta hai.
Us number ka sign batata hai ki shape flip hui ya nahi; uska size batata hai area stretch kitna hua. Aage padhne se pehle neeche ki picture zaroor dekho.

Red parallelogram woh shape hai jo unit square matrix ke act karne ke baad ban jaata hai. Uska area $∣ det (A) ∣$ ke barabar hai, aur agar corner-labels "ulti taraf" ghoomein (counterclockwise ki jagah clockwise), toh $det (A)$ negative hoga.

Scenario matrix

Har determinant problem jo tumhe kabhi bhi milegi woh in cells mein se kisi ek mein aati hai. Humara kaam hai neeche sab ko cover karna.

Cell	Kya cheez khaas banati hai	Example jo ise cover karta hai
C1 — plain positive	$a d - b c > 0$ , ordinary numbers	Ex 1
C2 — negative (flip)	$a d - b c < 0$ , orientation reverse ho jaata hai	Ex 2
C3 — zero / degenerate	columns parallel hain, area collapse ho jaata hai	Ex 3
C4 — one zero column	poora ek column $0$ hai	Ex 4
C5 — negatives inside	$a, b, c, d$ mein se kuch negative hain	Ex 5
C6 — fractions / decimals	non-integer entries	Ex 6
C7 — used in inverse	$det$ ko $A^{- 1}$ mein daalo	Ex 7
C8 — real-world word problem	ek zameen ke plot ka area	Ex 8
C9 — exam twist (unknown)	$k$ nikalo taaki $det = 0$ ho	Ex 9
C10 — limiting behaviour	kisi entry ko ek value ki taraf $\to$ jaane do, $det$ dekho	Ex 10

Worked examples

Forecast: andaza lagao — kya answer positive hoga, negative, ya zero? (Rows ek doosre ke multiples nahi lagte, toh shayad non-zero hoga.)

Falling diagonal: $a \cdot d = 4 \cdot 3 = 12$ . Yeh step kyun? Formula ka pehla piece hamesha top-left times bottom-right hota hai — yahi $a d - b c$ mein " $a d$ " hai.
Rising diagonal: $b \cdot c = 1 \cdot 2 = 2$ . Yeh step kyun? Yahi woh part hai jo hum subtract karte hain; yeh correct karta hai ki dono vectors kitna "lean" karte hain ek doosre ki taraf.
Subtract: $12 - 2 = 10$ .

$det (A) = 10$

Verify: Parallelogram ka area sides $(4, 2)$ aur $(1, 3)$ ke saath shoelace idea se $∣ a d - b c ∣ = ∣10∣ = 10$ . Positive $\Rightarrow$ orientation kept. ✔

Forecast: dekho yeh Example 1 hai jisme dono columns swap hain. Columns swap karne se sign flip hota hai, toh predict karo $- 10$ .

$a d = 1 \cdot 2 = 2$ .
$b c = 4 \cdot 3 = 12$ .
$a d - b c = 2 - 12 = - 10$ .

$det (A) = - 10$

Negative kyun matter karta hai: minus sign kehta hai ki transformed square andar-bahar ho gaya (jaise pancake palat diya ho). Dono figures dekho — corner arrows opposite direction mein ghoomte hain.

Verify: Column swap ko Example-1 ke determinant ko $- 1$ se multiply karna chahiye: $- 1 \times 10 = - 10$ . ✔ Size $∣ - 10∣ = 10$ (Ex 1 jaisa hi area stretch, jaisa expect tha). ✔

Forecast: doosra column $(6, 9)$ lagta hai $3 \times (2, 3)$ — pehla column. Parallel columns $\Rightarrow$ predict karo $0$ .

$a d = 2 \cdot 9 = 18$ .
$b c = 6 \cdot 3 = 18$ .
$18 - 18 = 0$ .

$det (A) = 0$

Kaisa dikhta hai: dono vectors ek hi direction mein point karte hain, toh "parallelogram" ki width zero ho jaati hai — yeh ek line par squeeze ho jaata hai. Koi area nahi, koi inverse nahi.

Verify: Kya column 2, column 1 ka multiple hai? $(6, 9) = 3 \cdot (2, 3)$ . Haan $\Rightarrow$ dependent columns $\Rightarrow$ determinant $0$ hona hi chahiye. ✔

Forecast: poora pehla column zero vector hai. Zero-length side area enclose nahi kar sakta, toh predict karo $0$ .

$a d = 0 \cdot 7 = 0$ .
$b c = 5 \cdot 0 = 0$ .
$0 - 0 = 0$ .

$det (A) = 0$

Yeh phir bhi "degenerate" kyun hai: zero column dependency ka extreme case hai — parallelogram ki ek side collapse ho ke ek point ban gayi. Sab kuch vertical line $x = 0$ par map ho jaata hai.

Verify: Kisi bhi matrix mein zero column ho toh determinant $0$ hota hai; yahan dono products zero hain. ✔

Forecast: andar negatives hona theek hai — bas carefully plug in karo. Do negatives multiply ho ke positive bante hain.

$a d = (- 3) \cdot (- 5) = 15$ . Yeh step kyun? Sign rule: negative $\times$ negative $=$ positive. Yahi jagah hai jahan careless students galti karte hain.
$b c = 2 \cdot 4 = 8$ .
$15 - 8 = 7$ .

$det (A) = 7$

Verify: Sign ke saath dubara compute karo: $(- 3) (- 5) - (2) (4) = 15 - 8 = 7 > 0$ , toh orientation preserved hai chahe entries mein minus signs hon. ✔

Forecast: fractions method nahi badlate; expect karo ek chota sa number.

$a d = 0.5 \cdot 3 = 1.5$ .
$b c = 2 \cdot 1 = 2$ .
$1.5 - 2 = - 0.5$ .

$det (A) = - 0.5$

Verify: $∣ - 0.5 ∣ = 0.5$ matlab shape ka area adha ho gaya, aur negative sign orientation flip karta hai. Fraction mein convert karo: $- \frac{1}{2}$ . ✔

Forecast: inverse formula $det (A)$ se divide karta hai, toh pehle humein woh number non-zero chahiye.

$det (A) = (2) (3) - (5) (1) = 6 - 5 = 1$ . Yeh step kyun? Inverse of a 2×2 matrix se, inverse tabhi exist karta hai jab $det \neq = 0$ . Yahan $det = 1$ hai, toh safe hain.
Adjugate banao: $a \leftrightarrow d$ swap karo, $b$ aur $c$ ko negate karo: $[3 - 1 - 5 2]$
$det = 1$ se divide karo: $A^{- 1} = \frac{1}{1} [3 - 1 - 5 2] = [3 - 1 - 5 2]$

Verify: $A A^{- 1}$ multiply karo (dekho Matrix multiplication): identity $[1001]$ milna chahiye. ✔

Forecast: $O$ se do edge-vectors ek parallelogram span karte hain; triangle uska aadha hota hai. Guess karo area kaafi square metres mein hoga.

Edge vectors: $O P = (6, 1)$ , $O Q = (2, 5)$ . Inhe columns ki tarah rakho: $M = [6125]$ Yeh step kyun? Area of parallelogram us matrix ke $∣ det ∣$ ke barabar hota hai jiske columns dono spanning vectors hain.
$det (M) = (6) (5) - (2) (1) = 30 - 2 = 28$ .
Parallelogram area $= ∣28∣ = 28 m^{2}$ ; triangle aadha hai: $Area = \frac{1}{2} \cdot 28 = 14 m^{2}$

Verify: Triangle $O, P, Q$ par Shoelace: $\frac{1}{2} ∣ x_{P} y_{Q} - x_{Q} y_{P} ∣ = \frac{1}{2} ∣6 \cdot 5 - 2 \cdot 1∣ = \frac{1}{2} (28) = 14 m^{2}$ . Units area (m²) hain. ✔

Forecast: singular matlab $det = 0$ . Isse $k$ mein ek equation milegi — shayad do answers kyunki $k^{2}$ aayega.

Determinant likho: $det = k \cdot k - 4 \cdot 2 = k^{2} - 8$ .
Zero set karo: $k^{2} - 8 = 0$ . Yeh step kyun? "Singular" exactly $det = 0$ ka statement hai (koi inverse nahi, columns dependent).
Solve karo: $k^{2} = 8 \Rightarrow k = \pm 22$ .

$k = 22 or k = - 22$

Verify: $k = 22$ plug karo: $det = (22)^{2} - 8 = 8 - 8 = 0$ . ✔ $- 22$ ke liye bhi same. ✔

Forecast: $t = 0$ par yeh identity hai ( $det = 1$ ); jaise off-diagonal badhta hai, columns ek doosre ki taraf jhukne lagte hain, toh $det$ shrink hona chahiye, kahin zero hit karna chahiye, phir negative ho jaana chahiye.

Determinant: $det = (1) (1) - (t) (t) = 1 - t^{2}$ .
Boundary/limiting cases evaluate karo:
- $t = 0$ : $1 - 0 = 1$ (identity, koi distortion nahi).
- $t = 1$ : $1 - 1 = 0$ (columns identical ho jaate hain $\Rightarrow$ ek line par collapse).
- $t = 2$ : $1 - 4 = - 3$ (columns cross over ho gaye $\Rightarrow$ orientation flip). Yeh step kyun? Ek entry ko sweep karne se hum determinant ko positive → zero → negative se guzarte dekh sakte hain — ek hi family mein teeno cases.

$det (A (t)) = 1 - t^{2}$

Verify: $t = 0, 1, 2$ par values $1, 0, - 3$ hain, jo sign story se match karte hain. ✔

Recall Quick self-test

Example 2, Example 1 ka negative kyun hai? ::: Dono same matrix hain bas columns swapped hain, aur column swap determinant ko $- 1$ se multiply karta hai. Example 8 mein 2 se kyun divide kiya? ::: Determinant parallelogram ka area deta hai; triangle exactly uska aadha hota hai. Example 10 mein $t \geq 0$ ke liye kis $t$ par matrix non-invertible ho jaati hai? ::: $t = 1$ par, jahan $det = 0$ hai.

Connections

Inverse of a 2×2 matrix — Example 7 mein invert karne se pehle $det$ use hota hai.
Area of parallelogram — Example 8 ka word problem.
Linear independence — Examples 3, 4, 9 sab dependent columns par hinge karte hain.
Cramer's rule — $det \neq = 0$ chahiye, same test jaisa Example 9 mein hai.
Transformations and scaling — Example 2 mein flip aur Example 10 mein collapse.