Determinant of 2×2 matrix

2.6.7 · Maths › Matrices & Determinants — Introduction

HUM kyun care karte hain? Kyunki determinant humein invertibility bataata hai (non-zero ⇒ invertible), solution existence (Cramer's rule), aur geometric transformations ke baare mein.

Definition & Formula

First Principles se Derivation:

YEH formula kyun? Chaliye isse geometric interpretation se derive karte hain.

Step 1: Unit square ko consider karo jiske corners $(0,0),(1,0),(0,1),(1,1)$ par hain.

Step 2: Basis vectors par transformation $A$ apply karo:

• $\left[\begin{array}{c}1\\ 0\end{array}\right]\to \left[\begin{array}{c}a\\ c\end{array}\right]$ (pehla column)
• $\left[\begin{array}{c}0\\ 1\end{array}\right]\to \left[\begin{array}{c}b\\ d\end{array}\right]$ (doosra column)

Step 3: Unit square ek parallelogram ban jaata hai jiske vertices origin, $(a,c)$, $(b,d)$, aur $(a+b,c+d)$ par hain.

Step 4: Vectors $\vec{u}=(a,c)$ aur $\vec{v}=(b,d)$ se bane parallelogram ki area cross product magnitude se milti hai (2D mein, $z$-component): $\text{Area}=\left|\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|\right|=|ad-cb|$

"$ad-bc$" specifically kyun? Yeh Shoelace formula se aata hai polygon area ke liye:

• Positive contributions: $a\cdot d$ (diagonal product)
• Negative contributions: $b\cdot c$ (anti-diagonal product)
• Sign orientation capture karta hai (counterclockwise = positive, clockwise = negative)

Memory trick: "Down-right minus up-right" — main diagonal multiply karo, off-diagonal product subtract karo.

Worked Examples

Solution: $\det (A)=(3)(4)-(5)(2)=12-10=2$

Yeh step kyun? Hum formula directly apply kar rahe hain: main diagonal product minus off-diagonal product.

Interpretation: Yeh transformation areas ko 2 ke factor se scale karta hai (areas double ho jaati hain), aur orientation preserve karta hai (positive determinant).

Solution: $\det (A)=(6)(2)-(3)(4)=12-12=0$

Zero kyun? Notice karo ki doosri row pehli row ki $\frac{2}{3}$ times hai: $[4,2]=\frac{2}{3}[6,3]$.

Geometric meaning: Dono column vectors parallel hain (ek doosre ka scalar multiple hai), isliye woh ek line span karte hain, area nahi. Yeh transformation 2D space ko ek 1D line par collapse kar deta hai.

Consequence: $A$ ka no inverse hai (singular matrix). Systems $Ax=b$ ke ya toh koi solution nahi hoga ya infinitely many solutions honge.

Solution: $\det (A)=(2)(1)-(5)(3)=2-15=-13$

Negative kyun? Yeh transformation orientation reverse karta hai — space ko flip karta hai jaise ek glove ko ulta karna.

Physical analogy: Agar tum unit square ke corners ko clockwise label karo, toh transformation ke baad woh counterclockwise ho jaate hain (ya vice versa). Absolute value $|-13|=13$ ka matlab hai ki areas 13 ke factor se scale hoti hain.

Solution: $\det (I)=(1)(1)-(0)(0)=1$

1 kyun? Identity matrix kuch nahi badlaata — yeh har vector ko khud par map karta hai. Areas scale nahi hoti, isliye determinant 1 hai (multiplicative identity).

General principle: Kisi bhi invertible matrix ke liye, $\det (A)\cdot \det (A^{-1})=\det (I)=1$. Toh $\det (A^{-1})=\frac{1}{\det (A)}$.

Common Mistakes

Yeh sahi kyun lagta hai: Students "products subtract karo" toh yaad rakhte hain par bhool jaate hain ki pehle kaunsa aata hai.

Fix: Hamesha main diagonal PEHLE karo (top-left se bottom-right yaani $ad$), phir off-diagonal subtract karo ($bc$). Mnemonic: "Diagonal Down first" (D up-diagonal se pehle aata hai).

Steel-man: Wrong formula $bc-ad=-(ad-bc)$ sahi answer ka negative deta hai. Yeh determinant ko reversed orientation ke saath compute kar raha hai. Dono scaling factor magnitude capture karte hain, par sign direction ke liye matter karta hai.

Yeh sahi kyun lagta hai: Trace $\text{tr}(A)=a+d$ bhi ek important matrix quantity hai, aur dono matrix se compute hone wale single numbers hain.

Fix:

• Trace (sum): $a+d$ — "basis directions ke along total stretch" measure karta hai
• Determinant (product minus product): $ad-bc$ — "area scaling" measure karta hai

Distinguish karne ka example: $\left[\begin{array}{cc}2& 0\\ 0& 3\end{array}\right]$ ka trace $=5$ hai, determinant $=6$ hai. Diagonal matrices mein fark clearly dikhta hai: trace add karta hai, determinant diagonal entries ko multiply karta hai.

Yeh sahi kyun lagta hai: Zero ko "kuch nahi" se associate kiya jaata hai.

Fix: $\det (A)=0$ ka matlab hai ki matrix singular (non-invertible) hai — iske columns linearly dependent hain. Bahut saari non-zero matrices ka determinant zero hota hai: $\left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right],\left[\begin{array}{cc}5& 10\\ 1& 2\end{array}\right],\left[\begin{array}{cc}0& 1\\ 0& 2\end{array}\right]$

Key insight: Ek column doosre ka multiple hai, YA ek column zero hai (dependency ka special case).

Properties & Key Insights

1. Multiplicativity: $\det (AB)=\det (A)\cdot \det (B)$ — jab matrices multiply hoti hain toh determinants bhi multiply hote hain.

2. Transpose invariance: $\det (A^T)=\det (A)$ — transpose ke under determinant nahi badlaata.

3. Inverse formula: Agar $\det (A)\ne 0$ hai, toh: $A^{-1}=\frac{1}{\det (A)}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$ KYU? Yeh $A\cdot A^{-1}=I$ se aata hai. Adjugate matrix $a\leftrightarrow d$ swap karta hai, off-diagonals negate karta hai.

4. Row swap se sign flip: Rows (ya columns) swap karne se determinant $-1$ se multiply ho jaata hai.

Determinant tumhe do cheezein bataata hai:

1. Original square ke comparison mein naya shape kitna bada ya chhota hua (yeh number ki size hai)
2. Kya tumne use ulta kar diya? (yeh + ya − sign hai)

Ek 2×2 matrix ke liye, tum ise aise calculate karte ho: "Neechey girti" diagonal ke dono numbers lo (top-left aur bottom-right), unhe multiply karo. Phir "Upar chadhti" diagonal lo (bottom-left aur top-right), unhe multiply karo. Doosre ko pehle se subtract karo.

Agar tumhe zero milta hai, iska matlab hai tum square ko puri tarah flat karke ek line mein squeeze kar diya — ab uski koi area nahi hai! Yeh equations solve karne ke liye bura hai kyunki information kho gayi.

Agar tumhe ek negative number milta hai, tumne square ko ulta kar diya (pancake flip karne jaisa). Size abhi bhi minus ke bina wala number hai, par orientation change ho gayi.

Bas yehi hai! Determinant = "kitna area change + kya flip hua?"

Visual: Matrix ke through ek X draw karo. "" diagonal (girti hui) positive hai, "/" diagonal (chadhti hui) negative hai.

Connections

• Matrix multiplication — determinants multiply hote hain: $\det (AB)=\det (A)\det (B)$
• Inverse of a 2×2 matrix — exist karta hai sirf jab $\det (A)\ne 0$
• Linear independence — columns independent ⟺ $\det (A)\ne 0$
• Cramer's rule — $Ax=b$ solve karne ke liye determinants use karta hai
• Area of parallelogram — determinant ki geometric interpretation
• Eigenvalues — eigenvalues ka product determinant ke barabar hota hai
• Cross product in 3D — ek 3×3 matrix ka determinant
• Transformations and scaling — area/volume scaling factor ke roop mein determinant

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