Matrix multiplication — conditions, process, non-commutativity: Question bank

2.6.5 · D5 · Maths › Matrices & Determinants — Introduction › Matrix multiplication — conditions, process, non-commutativi

Yeh page tumhari samajh ki workout ke liye hai, arithmetic ke liye nahi. Neeche diya hua har item ek aisa jagah hai jahan students confidently galat cheez likhte hain. Question padho, answer chhupao, apni wajah zor se bolo, phir reveal karo. Agar tumhari wajah di gayi reasoning se match nahi karti, to yeh ek trap tha jisse tum abhi bache.

Prerequisites jo tumhare paas pehle se hone chahiye: parent note (dot-product rule aur dimension-matching condition), plus Linear transformations ka ek feel as the meaning behind the numbers.

Sach ya jhooth — justify karo

Agar

A B

defined hai, to

B A

bhi defined hai.

Jhooth.

A B

ke liye chahiye columns-of-

A

= rows-of-

B

;

B A

ke liye chahiye columns-of-

B

= rows-of-

A

— yeh bilkul alag condition hai. Jaise

A

hai

2 \times 3

B

hai

3 \times 4

A B

kaam karta hai,

B A

nahi.

Agar dono

A B

aur

B A

defined hain, to unka shape ek jaisa hona chahiye.

Saamanya taur par jhooth.

A_{2 \times 3}

aur

B_{3 \times 2}

A B

ka shape

2 \times 2

milta hai lekin

B A

ka shape

3 \times 3

. Same shape tabhi zaruri hai jab dono same size ke square hon.

Same size ke square matrices ke liye,

A B = B A

hamesha hota hai.

Jhooth. Dono products exist karte hain aur ek jaisa shape share karte hain, lekin woh aam taur par alag hote hain — parent note mein swap-matrix example dekho jahan

A B = [2413]

lekin

B A = [3142]

. Multiplication ka dono taraf defined hona iska matlab yeh nahi ki woh commute karta hai.

Matrix multiplication associative hai:

(A B) C = A (B C)

Sach. Yeh function composition ko mirror karta hai, jo associative hoti hai —

C

phir

B

phir

A

lagana, kisi bhi grouping mein same overall transformation deta hai.

Matrix multiplication entry-by-entry hoti hai, jaise addition.

Jhooth. Addition entry-wise hoti hai, lekin multiplication ek row-column dot product hai. Entry-wise version ek alag operation hai (Hadamard product

A \circ B

Agar

A B = O

(zero matrix) hai, to

A

B

mein se kam se kam ek zero hona chahiye.

Jhooth. Non-zero matrices multiply hokar zero de sakti hain, jaise

[1000] [0001] = [0000]

. Matrices mein "zero divisors" hote hain — numbers mein nahi hote.

Agar

A B = A C

hai, to

B = C

hoga.

Jhooth. Tum "

A

" cancel nahi kar sakte jab tak

A

invertible na ho. Agar

A

space ko collapse karta hai (uske paas inverse nahi hai), to alag

B

aur

C

same output pe land kar sakte hain.

(A B)^{T} = A^{T} B^{T}

Jhooth — order ulta ho jaata hai:

(A B)^{T} = B^{T} A^{T}

. Matrix transpose properties dekho; transpose karna composition order ko flip karta hai, bilkul jaise socks-then-shoes ko undone karna.

det (A B) = det (A) det (B)

Sach, aur notable yeh hai ki

det (A B) = det (B A)

even when

A B \neq = B A

. Determinant area/volume scaling measure karta hai, aur total scaling ko order ki parwah nahi.

Identity matrix se multiply karna ek matrix ko badal sakta hai.

Jhooth.

A I = I A = A

; identity "do-nothing" transformation hai, isliye yeh har matrix ko dono sides se untouched chhod deta hai.

Galti dhundho

A

hai

2 \times 2

B

hai

3 \times 1

, to

A B

hai

2 \times 1

." Galti nikalo.

Inner dimensions pehle match hone chahiye:

A

2

columns vs

B

3

rows (

2 \neq = 3

), isliye

A B

undefined hai — tum shape tak pahunchte hi nahi.

C_{21}

compute karne ke liye,

A

ka column 2,

B

ke row 1 ke saath dot karo." Galti nikalo.

Ulta hai.

C_{ij}

mein ==row

i

A

aur column

j

B

== use hoti hai, isliye

C_{21}

mein

A

ki row 2 aur

B

ka column 1 use hoga.

A B

exist karta hai aur

B A

exist karta hai, to multiplication yahan commutative hai." Galti nikalo.

Dono products ka exist karna ek dimension fact hai; commutativity ek value fact hai. Tumhe actually dono compute karke compare karne honge — woh rarely agree karte hain.

"Kyunki

2 \times 3

times

3 \times 2

2 \times 2

milta hai, aur

3 \times 2

times

2 \times 3

3 \times 3

milta hai, dono identity-sized answer hain." Galti nikalo.

Dono automatically identity nahi hain; woh sirf alag shapes ke valid products hain. Shape ≠ content, aur square hona tumhe

I

nahi banata.

"Main factor karta hun:

A B + A C = (A + A) (B + C)

." Galti nikalo.

Sahi distributive law hai

A (B + C) = A B + A C

; tum common

A

factor out karte ho, tum

A

's add nahi karte.

(A + A) (B + C) = 2 A (B + C)

ek bilkul alag quantity hai.

(A + B)^{2} = A^{2} + 2 A B + B^{2}

." Galti nikalo.

Expand karne par milta hai

A^{2} + A B + B A + B^{2}

, aur kyunki saamanya taur par

A B \neq = B A

hai, tum unhe

2 A B

mein merge nahi kar sakte. Binomial shortcut commutativity assume karta hai.

Why questions

(i, j)

entry ek row aur column ka dot product kyun hai, do numbers ka product kyun nahi?

Kyunki

B

ka column

j

batata hai ki ek basis vector kahan jaata hai, aur

A

ki row

i

batati hai ki result ke ek coordinate ko kaise read karna hai —

k

ke upar

A_{ik} B_{k j}

ka sum karna saare intermediate dimensions ko ek output coordinate mein blend karta hai.

Inner dimensions match kyun hone chahiye?

A

ki rows

n

components wala input expect karti hain;

B

ke columns exactly

n

components supply karne chahiye. Agar

B

ke columns ki alag height ho, to dot karne ke liye kuch consistent nahi hai.

Order kyun matter karta hai —

A B

generally

\neq = B A

kyun hota hai?

Kyunki

A B

ka matlab hai "pehle

B

karo, phir

A

", aur transformations ko opposite order mein compose karna vectors ko doosri jagah le jaata hai — rotate-then-stretch, stretch-then-rotate se alag hota hai.

det (A B) = det (B A)

kyun hai even though

A B \neq = B A

Determinant ek scalar hai jo area/volume kitna scale hota hai yeh record karta hai. Dono orders same do scalings apply karte hain, aur

det (A) det (B) = det (B) det (A)

kyunki numbers commute karte hain — "total stretch" ki geometry order-free hai.

(A B)^{T} = B^{T} A^{T}

mein order kyun ulta ho jaata hai?

Transpose karna rows aur columns ki roles swap karta hai, jo exactly "undo in reverse" pattern hai — jo last operation apply ki gayi thi woh transpose hone wali pehli operation ban jaati hai.

Hum

A B = A C

A

generally cancel kyun nahi kar sakte?

Cancel karna secretly

A^{- 1}

se multiply karna hai, lekin agar

A

ek dimension squash kar deta hai to uska inverse nahi hai. Ek collapsing map alag

B, C

ko same image pe bhej sakta hai, isliye cancellation illegal hai.

Edge cases

A \cdot O

kya hai jahan

O

compatible shape ki zero matrix hai?

Zero matrix. Har entry ek all-zero column ke saath dot product hai, isliye har sum

0

hota hai — transformation sab kuch origin pe bhej deta hai.

Jab tum

1 \times n

row ko

n \times 1

column se multiply karte ho to kya hota hai?

Tumhe

1 \times 1

matrix milti hai — ek single number, do vectors ka plain dot product. Yeh sabse chhota non-trivial matrix product hai.

Jab tum

n \times 1

column ko

1 \times n

row se multiply karte ho (reverse order) to kya hota hai?

Tumhe

n \times n

matrix milti hai (ek "outer product"), scalar nahi. Same do vectors, opposite order, wildly different shape — ek vivid reminder ki order shape tak ko control karta hai.

Kya ek

1 \times 1

matrix times ek

1 \times 1

matrix sirf ordinary number multiplication hai?

Haan — dot-product rule ek single term

A_{11} B_{11}

tak collapse ho jaata hai, isliye matrix multiplication apne sabse chhote case mein ordinary multiplication contain karta hai.

Kya ek valid matrix

A

ke liye

A^{2}

undefined ho sakta hai?

Haan, agar

A

non-square ho:

A^{2} = A \cdot A

ke liye columns-of-

A

= rows-of-

A

chahiye, jo tabhi hold karta hai jab

A

square ho. Powers ek square-matrix privilege hai.

Agar

D

diagonal hai, to kya

D A

A

ki rows scale karta hai ya columns?

D A

A

ki rows scale karta hai (har row

i

D_{ii}

se), jabki

A D

columns scale karta hai — ek clean case jahan tum clearly dekh sakte ho kyun left- aur right-multiplication alag hoti hai.

Recall Yahan har trap ki ek-line summary

Shape rules (inner match, order changes shape), value rules ( $A B \neq = B A$ , no cancelling, zero divisors), aur algebra rules ( $(A B)^{T} = B^{T} A^{T}$ , $A (B + C) = A B + A C$ ) — sab ek hi idea par trace back hote hain: matrices ordered compositions of transformations hain, numbers nahi.