Matrix operations — addition, subtraction (conditions): Visual walkthrough

2.6.3 · D2 · Maths › Matrices & Determinants — Introduction › Matrix operations — addition, subtraction (conditions)

Yeh page matrix addition ko bilkul scratch se rebuild karta hai. Koi symbol tab tak nahi aayega jab tak hum use draw nahi kar lete. Akhir mein tum dekh paoge ki do matrices ko tab hi add kiya ja sakta hai jab unki shape same ho — yeh rule kisi ne thuma hua nahi lagega, balki obvious lagega.

Hum Matrix Notation and Terminology pe build kar rahe hain "row", "column", aur "order" jaise words ke liye, aur raaste mein Zero Matrix and Identity Matrix, Scalar Multiplication of Matrices, Transpose of a Matrix aur the parent ki taraf point karte hain.

Step 1 — Matrix actually hai kya: ek labelled grid

KYA HAI. Ek matrix bas ek rectangle of numbers hai jo rows (horizontal lines) aur columns (vertical lines) mein arranged hoti hai. Bas itna hi.

KYUN. Do grids ko combine karne se pehle humein agree karna hoga ki ek grid ka matlab kya hai. Har number ek box mein baitha hai, aur us box ki position ka matlab hota hai — "row 1, column 2" ek alag slot hai "row 2, column 1" se, chahe numbers equal hi kyun na hon.

PICTURE. Neeche di gayi grid dekho. Magenta mein highlighted box row 2, column 3 pe hai. Hum iska address $a_{23}$ likhte hain — padho "a-two-three". Chota " $2$ " batata hai kaunsi row, " $3$ " batata hai kaunsa column. Chota letter $a$ bas yeh batata hai ki yeh number kis matrix ka hai.

Step 2 — "Order" grid ki shape hai

KYA HAI. Ek matrix ka order $m \times n$ likha jaata hai, bola jaata hai " $m$ by $n$ ". Yahan $m$ rows ki sankhya hai aur $n$ columns ki sankhya hai.

KYUN. Do grids mein numbers ki count same ho sakti hai lekin unki shapes bilkul alag ho sakti hain. Grids ko compare karne ke liye humein unki shape ki baat karni hogi, total count ki nahi. Order hi woh shape hai.

PICTURE. Baayeen grid $2 \times 3$ hai — 2 rows, 3 columns, 6 boxes. Daayeen grid $3 \times 2$ hai — 3 rows, 2 columns, fir bhi 6 boxes. Boxes ki sankhya same, shape alag. Arrows trace karo: box "row 1, column 3" baayeen taraf exist karta hai lekin daayeen taraf column 3 hai hi nahi.

Step 3 — Add karne ka matlab hai same address pe boxes ko pair karna

KYA HAI. Do matrices $A$ aur $B$ ko add karne ke liye hum har address $(i, j)$ pe jaate hain aur wahan baithein dono numbers ko add karte hain: $c_{ij} = box (i, j) of A a_{ij} + box (i, j) of B b_{ij}$

Term by term padho: output box $c_{ij}$ ko $A$ ke us address wale box ka number milta hai plus $B$ ke exactly same address wale box ka number. Same row, same column, dono baar.

KYUN. Har box ka kuch specific matlab hota hai ("Monday–dishes minutes"). Ek chart ke box $(1, 1)$ ko doosre chart ke box $(1, 1)$ se add karna meaning ko intact rakhta hai: hum do sources se same fact ko combine kar rahe hain. Isliye is operation ko element-wise kehte hain — box apne twin se milta hai.

PICTURE. Do transparent grids ek doosre ke upar stack ki hui hain. Har box wahan glow karta hai jahan uska partner seedha neeche baitha hai; hum sum us slot mein daal dete hain. Har add strictly vertical hai — box $(i, j)$ sirf box $(i, j)$ se milta hai.

Step 4 — Ek worked add, box by box

KYA HAI. Lo

B = \begin{bmatrix} 4 & 1 & -1 \\ -3 & 7 & 6 \end{bmatrix}.$$ Dono $2\times 3$ hain, toh har box ka ek partner hai. Har pair ko add karo: $$A+B = \begin{bmatrix} 2+4 & -3+1 & 5+(-1) \\ 1+(-3) & 0+7 & -2+6 \end{bmatrix} = \begin{bmatrix} 6 & -2 & 4 \\ -2 & 7 & 4 \end{bmatrix}.$$ **KYUN.** Humne pehle *shape check ki* ($2\times3 = 2\times3$ ✓), kyunki tabhi har box ka twin hoga. Phir humne chhe independent additions kiye — har address pe ek — koi box akela nahi raha. **PICTURE.** Har output box coloured hai, uske do source boxes arrows se trace kiye gaye hain. Box $(1,2)$ dekho: usne $A$ se $-3$ aur $B$ se $+1$ liya aur $-2$ banaya. Kuch bhi columns ya rows cross nahi karta. Khud se verify karo: - Box $(2,1)$ of $A+B$ ::: $1+(-3) = -2$ - Box $(1,3)$ of $A+B$ ::: $5+(-1) = 4$ --- ## Step 5 — Subtraction same picture hai, bas ek flip ke saath **KYA HAI.** Subtraction bas har box ka **negative** add karta hai: $$d_{ij} = a_{ij} - b_{ij} = a_{ij} + (-b_{ij}).$$ Yahan $-b_{ij}$ wahi number hai jo $b_{ij}$ hai lekin sign flip hua hai. Yeh [[Scalar Multiplication of Matrices]] se jud ta hai: $-B$ matlab $B$ ko $-1$ se scale kiya. **KYUN.** "Minus" ko "negative add karo" mein badal kar, subtraction *exactly same* twin-pairing machinery inherit karta hai — isliye isko *exactly same* matching shapes chahiye. Hum koi nayi rule nahi banate. **PICTURE.** Step 3 wali same stacked grids, lekin $B$ ki layer "negated" dikhaayi gayi hai (signs flip, violet mein draw ki gayi). Phir pehle ki tarah add karte hain. Quick worked case: $$P=\begin{bmatrix}5&-1\\3&2\\0&4\end{bmatrix},\quad Q=\begin{bmatrix}2&3\\-1&1\\5&-2\end{bmatrix} \;\Rightarrow\; P-Q=\begin{bmatrix}5-2&-1-3\\3-(-1)&2-1\\0-5&4-(-2)\end{bmatrix} =\begin{bmatrix}3&-4\\4&1\\-5&6\end{bmatrix}.$$ Dono $3\times2$ hain — twins har jagah exist karte hain, toh defined hai. --- ## Step 6 — Degenerate case: shapes agree nahi karti **KYA HAI.** Add karne ki koshish karo: $$M = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}\;(2\times 3),\qquad N = \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix}\;(3\times 2).$$ **KYUN.** Address $(1,3)$ pe jaao: $M$ mein wahan box hai (number $3$), lekin $N$ mein **column 3 hai hi nahi** — uska box missing hai. $3$ ko add karne ke liye kuch nahi. Address $(3,1)$ pe jaao: $N$ mein box hai ($11$), lekin $M$ mein **row 3 hai hi nahi**. Do orphans, koi twin nahi. **PICTURE.** Orphaned boxes orange ring mein hain aur "?" ke saath — woh bina kisi partner ke float kar rahe hain. Kyunki algebra deterministic hona chahiye, hum ek value **invent nahi kar sakte**, isliye poora operation ==undefined== declare ho jaata hai. > [!mistake] "Same total count kaafi hai" > $M$ aur $N$ dono mein 6 numbers hain, toh *lagta hai* add ho sakta hai. Lekin addition pairs by **address** karta hai, count se nahi. Lalchaane wala fix "bas $N$ ko transpose kar do" ek *alag* operation deta hai $M + N^{\mathsf T}$ (dekho [[Transpose of a Matrix]]) — yeh har box ka matlab badal deta hai, isliye yeh $M$ aur $N$ ka sum nahi hai. > [!recall]- Har shape case ek nazar mein > Same order → defined ::: haan, har twin ko add karo > Different order (chahe total count same ho) → defined ::: nahi, undefined hai > Same order ka zero matrix $O$ add karna → result ::: unchanged, $A+O=A$ (dekho [[Zero Matrix and Identity Matrix]]) --- ## Ek-picture summary Upar sab kuch ek flow mein collapse ho jaata hai: **shape pe agree karo → twin boxes pair karo → add karo → same address mein daal do.** Agar kisi box ka twin nahi, ruko — undefined. > [!recall]- Feynman retelling > Tum aur tumhara dost dono apna-apna chore chart rakhte ho: upar ki taraf days, side mein chores, aur andar minutes. Apna *combined* effort nikaalene ke liye tum square by square jaate ho: "Monday–dishes: maine 15 kiye, tune 20 kiye, saath mein 35." Tum Monday–dishes ko Tuesday–laundry se kabhi nahi milaate — sirf *same square* *same square* se milta hai. Yeh tabhi kaam karta hai jab dono charts ka layout identical ho: same days across, same chores down. Agar tumhare chart mein ek extra day hai jo dost ke chart mein nahi hai, toh us column ka koi partner square nahi, isliye poora combining impossible hai — undefined. Subtraction wahi walk hai, bas pehle dost ke numbers negative kar do ("tune mujhse 5 minute zyada kiye"). Yahi poori kahaani hai matrix addition aur subtraction ki: same shape, pair the twins, done. --- ## Connections - [[Matrix Operations — Addition, Subtraction (Conditions)]] — parent rule jo is page ne derive ki. - [[Matrix Notation and Terminology]] — jahan se "row", "column", "order", $a_{ij}$ aate hain. - [[Zero Matrix and Identity Matrix]] — addition ke liye identity $O$. - [[Scalar Multiplication of Matrices]] — $-B$ matlab $B$ ko $-1$ se scale kiya; subtraction ko power karta hai. - [[Transpose of a Matrix]] — kyun $A+B^{\mathsf T}$ ek *alag* operation hai. - [[Matrix Multiplication]] — contrast: multiplication ko $n$ (of $A$) $=$ rows (of $B$) chahiye, equal shapes nahi. - [[System of Linear Equations]] — systems ko combine karna matlab matrices ko combine karna.