Matrix operations — addition, subtraction (conditions): Foundations

2.6.3 · D1 · Maths › Matrices & Determinants — Introduction › Matrix operations — addition, subtraction (conditions)

Matrices add karne se pehle, tumhe us language mein fluent hona padega jo parent page bolti hai. Neeche har ek symbol aur idea hai jo wo use karta hai, bilkul zero se banaya gaya hai, har ek apne pehle wale par tika hua hai.

1. Matrix hoti kya hai — the grid

Figure dekho. Ek row left-to-right (horizontal) chalti hai. Ek column top-to-bottom (vertical) chalta hai. Poori cheez ek object hai jise hum ek single capital-letter naam dete hain, jaise $A$ .

Ye topic ko kyun chahiye: tum "do cheezon ko add" nahi kar sakte jab tak tum jaante nahi ki ek cheez kya hai. Grid object hai; addition ek action hai grids par.

Poori vocabulary tour ke liye dekho Matrix Notation and Terminology.

2. Rows, columns, aur ek number ka "address"

Grid ke andar har number ek box mein baith ta hai, aur us box ka ek address hota hai: kaun si row, phir kaun sa column.

Ye topic ko kyun chahiye: parent page ka poora argument — "same logical slot wali entries add karo" — bekar hai jab tak "slot" (= address) bilkul clear na ho. Do numbers tabhi add ho sakte hain jab unka address same ho.

3. Symbol $a_{ij}$ — subscript padhna

Har baar "matrix $A$ ki row 2, column 3 mein rehne wala number" likhna bahut thaka dene wala hai. Mathematicians ise $a_{23}$ tak compress kar dete hain.

Ye topic ko kyun chahiye: addition rule literally $c_{ij} = a_{ij} + b_{ij}$ likha hota hai. Ye jaane bina ki $i$ row hai aur $j$ column hai, wo line gibberish hai.

4. Order / dimensions — shape $m \times n$

Matrix ki shape do counts se describe hoti hai.

Ye topic ko kyun chahiye: matrices add karne ki ek hi condition hai "orders identical hone chahiye". Order wo yardstick hai jo decide karta hai ki do grids mil sakti hain ya nahi.

5. Bracket-notation shorthand $A = [a_{ij}]_{m \times n}$

Parent page poori grid ko $A = [a_{ij}]_{m \times n}$ ke roop mein compactly likhta hai. Ab har piece earned hai:

Ye topic ko kyun chahiye: ye compressed sentence hai jis par definition aur derivation sections rely karte hain. Ye hume ek generic entry $a_{ij}$ ke baare mein baat karke saari entries ke baare mein ek saath baat karne deta hai.

6. $+$ aur $-$ signs — aur "element-wise"

Tum pehle se ordinary numbers ke liye $+$ aur $-$ jaante ho. Naya phrase hai element-wise.

$c_{ij} = a_{ij} + b_{ij} for every address (i, j)$

Ye topic ko kyun chahiye: "element-wise" addition aur subtraction dono ka engine hai. Isliye matching shapes matter karti hain — ek grid ke har box ko doosri grid mein apna partner box milna chahiye.

7. Special grids jo tumhe milenge: zero matrix $O$

Ye topic ko kyun chahiye: parent page ki identity property kehti hai $A + O = A$ — zeros add karne se kuch nahi badalta, box by box. Poori detail Zero Matrix and Identity Matrix mein hai.

8. Matrix ka negative, $- A$

Ye topic ko kyun chahiye: parent page subtraction ko $A - B = A + (- B)$ aur additive-inverse property ko $A + (- A) = O$ ke roop mein define karta hai. Dono ke liye pehle $- A$ define hona chahiye. Har entry ke signs flip karna exactly Scalar Multiplication of Matrices hai scalar $- 1$ ke saath.

9. Do ideas jo tum sunoge lekin yahan compute NAHI karoge

Ye parent page ke contrast notes mein aate hain, toh inhe pehchano — ye neighbors hain, wo tools nahi jo tum add karne ke liye use karte ho.

Transpose $B^{T}$ — ek grid ko flip karna taaki uski rows columns ban jayein (ek $m \times n$ ko $n \times m$ mein badal deta hai). Parent warn karta hai ki $A + B^{T} \neq = A + B$ . Dekho Transpose of a Matrix.
Matrix multiplication — ek alag operation hai ek alag shape rule ke saath ( $m \times n$ times $n \times p$ ). Iske rule ko addition rule ke saath confuse mat karo. Dekho Matrix Multiplication.

Prerequisite map

Ise top-down padho: grid boxes deti hai, boxes addresses dete hain, addresses symbol $a_{ij}$ deta hai, shapes order deti hain, aur same-shape rule ke saath milke element-wise addition ko power karte hain — parent topic.

Equipment checklist

Khud test karo — reveal karne se pehle jawab zor se bolo.

a_{ij}

mein, kaun sa subscript row hai?

Pehla wala,

i

("floor then door").

Order

2 \times 3

kya describe karta hai?

Ek shape — 2 rows aur 3 columns — number 6 nahi.

"Element-wise" ka matlab kya hai?

Operation box by box karo, sirf un do numbers ka use karke jo same address

(i, j)

share karte hain.

Zero matrix

O

kya hai?

Required shape ki ek grid jisme har entry

0

ke barabar hai.

- A

kya hai?

Grid

[- a_{ij}]

—

A

ki har entry ka sign flip hua hua.

Do matrices ko add karne ke liye unka same order kyun hona chahiye?

Kyunki addition entries ko address se pair karta hai; agar shapes alag hain, toh kuch boxes ka koi partner nahi hoga, isliye sum undefined hai.

Kya

A + B^{T}

aur

A + B

same hain?

Nahi — transposing se change ho jata hai ki entries kaun si relationships encode karti hain, isliye ye ek alag operation hai.

Kya

2 \times 3

ka matlab "2 ko 3 se multiply karo" hai?

Nahi — yahan

\times

ek "by" separator hai jo shape describe karta hai, koi product nahi.

Connections

Matrix operations — addition, subtraction (conditions) — wo parent jiske liye ye page tumhe prepare karta hai
Matrix Notation and Terminology — order, entries, aur $a_{ij}$ ka source
Zero Matrix and Identity Matrix — jahan se $O$ aata hai
Scalar Multiplication of Matrices — $- A$ scalar $- 1$ times $A$ hai
Transpose of a Matrix — neighbor $B^{T}$ jo $B$ ke same nahi hai
Matrix Multiplication — ek alag operation alag shape rule ke saath
System of Linear Equations — jahan ye grids pehli jagah se aati hain