Matrix operations — addition, subtraction (conditions): Question bank

2.6.3 · D5 · Maths › Matrices & Determinants — Introduction › Matrix operations — addition, subtraction (conditions)

Is bank mein misconceptions dhundhe jaate hain, arithmetic nahi. Har answer mein ek reason diya gaya hai, kabhi bhi akela "haan"/"nahi" nahi. Agar tum har answer ko zor se defend kar sako, to tumne sach mein matrix addition aur subtraction samajh li hai.

Shuru karne se pehle, woh tools yaad karo jo tumhare paas pehle se hone chahiye — Matrix Notation and Terminology se (order $m \times n$ ), Zero Matrix and Identity Matrix se (sabse-zero matrix $O$ ), Scalar Multiplication of Matrices se (har entry ko ek number $k$ se multiply karna) aur Transpose of a Matrix se (rows ko columns mein palat dena, likha jaata hai $B^{T}$ ).

True or false — justify

True or false: Agar

A

aur

B

mein total entries ki sankhya same ho, to

A + B

defined hai.

False. Ek

2 \times 3

aur ek

3 \times 2

matrix dono mein 6 entries hain, lekin position

(1, 3)

pehle mein exist karti hai doosre mein nahi — count, structure nahi hota.

True or false: Matrix addition commutative hai, yaani

A + B = B + A

jab bhi dono defined hon.

True. Har entry

a_{ij} + b_{ij} = b_{ij} + a_{ij}

follow karti hai kyunki real-number addition commutative hai, aur yeh slot-by-slot hold karta hai.

True or false: Matrix subtraction commutative hai, yaani

A - B = B - A

False. Subtraction anti-commutative hoti hai:

A - B = - (B - A)

, isliye jab tak

A = B

na ho, dono results har entry mein sign se alag honge.

True or false: Kisi bhi matrix

A

ke liye

A + O = A

, ek fixed zero matrix

O

use karke.

Jaisa likha hai, False —

O

ka order

A

ke saath same hona chahiye. Zero matrix koi ek object nahi hai; ek

2 \times 3

zero hota hai, ek

3 \times 3

zero hota hai, wagera.

True or false: Kisi bhi square matrix

A

ke liye,

A + A^{T}

aur

A - A^{T}

dono defined hain.

True. Ek square

n \times n

matrix ka

A^{T}

bhi

n \times n

hota hai, isliye orders match karte hain; gaur karo ki

A - A^{T}

ka diagonal sab-zero hoga kyunki

a_{ii} - a_{ii} = 0

True or false: Agar

A

2 \times 3

hai aur

B

3 \times 2

hai, to

A + B^{T}

ek valid matrix addition hai

A

aur

B

ka.

False.

A + B^{T}

ek legal operation hai (dono

2 \times 3

hain), lekin yeh

A

B

ke transpose ke saath add karta hai,

B

ke saath nahi —

B

ki entries relocate ho gayi hain, unka meaning badal gaya hai.

True or false: Addition associative hai:

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

jab sab same order ke hon.

True. Slot-by-slot,

(a_{ij} + b_{ij}) + c_{ij} = a_{ij} + (b_{ij} + c_{ij})

real-number associativity se, isliye grouping kabhi matter nahi karta.

True or false: Ek scalar

k

ke liye

k (A + B) = k A + k B

(Scalar Multiplication of Matrices dekho).

True. Har entry deta hai

k (a_{ij} + b_{ij}) = k a_{ij} + k b_{ij}

real numbers ke distributive law se.

True or false:

A

ka additive inverse

A^{T}

hai.

False. Additive inverse

- A = [- a_{ij}]

hai, jo har entry ko negate karke milta hai;

A + (- A) = O

. Transpose karna entries ko rearrange karta hai, unhe negate nahi karta.

Spot the error

Claim: "

A

3 \times 4

hai,

B

4 \times 3

hai; kyunki ek

3 \times 4

times

4 \times 3

allowed hai,

A + B

bhi allowed hai." — Kya galat hai?

Us insaan ne Matrix Multiplication ka rule (inner dimensions match) borrow kiya. Addition ko identical orders chahiye, isliye

3 \times 4

plus

4 \times 3

undefined hai.

Claim: "

[12] + [12] = [24]

." — Kya galat hai?

1 \times 2

row aur ek

2 \times 1

column ke alag-alag orders hain, isliye sum undefined hai; likhne wale ne silently ek ko reshape kar diya answer force karne ke liye.

Claim: "

A + B

jahan

A

2 \times 2

hai aur

B

2 \times 2

hai: maine sirf diagonals add kiye aur off-diagonals unchanged chhod diye." — Error?

Addition har slot ke liye element-wise hoti hai, sirf diagonal ke liye nahi — off-diagonal entries ko bhi

a_{ij} + b_{ij}

ki tarah sum karna zaroori hai.

Claim: "Kyunki

A - B

ko same order chahiye, main chhoti matrix ko zeros se pad karke phir bhi kar sakta hoon." — Error?

Padding karna aise entries invent karta hai jo original data mein the hi nahi, isliye deterministic definition toot jaati hai; operation undefined rehta hai, "fixable" nahi hota.

Claim: "

- A

matlab pehli entry ka sign palat do aur baaki chhod do." — Error?

- A = [- a_{ij}]

har entry ko negate karta hai, sirf ek ko nahi; tabhi

A + (- A) = O

har slot mein hold karta hai.

Claim: "Kyunki addition commutative hai,

A - B = B - A

." — Error?

Commutativity

+

ka property hai,

-

ka nahi. Subtraction

A + (- B)

ke roop mein define hoti hai, aur

A + (- B) \neq = B + (- A)

generally.

Why questions

Dono matrices ko dono dimensions kyun share karni chahiye, sirf ek nahi?

Har entry ko same

(i, j)

par ek partner chahiye; sirf rows match karne se columns unpaired reh jaate hain (ya vice-versa), isliye kuch slots mein add karne ke liye koi term hi nahi hota.

"Same total element count" ek tempting lekin galat condition kyun hai?

Total count yeh ignore karta hai ki kahan har number rehta hai; ek

2 \times 3

aur

3 \times 2

mein 6 numbers share hote hain lekin slot

(1, 3)

ek grid mein exist karta hai doosre mein nahi.

Subtraction addition se associativity aur identity kyun inherit karti hai lekin commutativity kyun kho deti hai?

Kyunki

A - B

ko define kiya jaata hai

A + (- B)

ke roop mein, isliye

+

ke saare properties carry over hote hain — sirf yeh exception hai ki operands swap karne se sign palat jaata hai, commutativity khatam ho jaati hai.

Zero matrix additive identity kyun hai, identity matrix

I

kyun nahi?

Addition ke liye hamein

a_{ij} + x_{ij} = a_{ij}

chahiye, jo force karta hai ki

x_{ij} = 0

har jagah ho;

I

ke diagonal par ones hote hain aur woh multiplication se belong karta hai, addition se nahi (Zero Matrix and Identity Matrix dekho).

B

ko add karne se pehle transpose karna meaning kyun change kar deta hai, chahe dimensions fix ho jaayein?

Transpose karna har entry ke row aur column role ko swap kar deta hai — agar

B

ka matlab tha "products × stores", to

B^{T}

ka matlab hai "stores × products", isliye tum alag-alag relationships combine kar rahe ho.

Matrix addition "do linear systems combine karna" kyun model kar sakta hai (System of Linear Equations se link)?

Kyunki same shape ke coefficient matrices slot-by-slot add hote hain, yeh mirror karta hai ki equation-by-equation tum same variables ke matching coefficients add karte ho.

Edge cases

Kya

A + A

defined hai, aur yeh kya hai?

Haan — same matrix, same order — aur

A + A = 2 A

hai, scalar-2 multiple, kyunki har slot deta hai

a_{ij} + a_{ij} = 2 a_{ij}

Kya

O + O

defined hai, aur result kya hai?

Haan, bas dono zero matrices ka order same hona chahiye; result phir se wahi same zero matrix hai, kyunki har slot mein

0 + 0 = 0

hota hai.

Kya do

1 \times 1

matrices add ho sakti hain, aur yeh ordinary numbers se kaise relate karti hai?

Haan; ek

1 \times 1

matrix

[a]

plus

[b]

hai

[a + b]

, isliye scalar addition matrix addition ka sabse chhota case hi hai.

Kya

A - A

defined hai, aur yeh kya hai?

Haan; har slot deta hai

a_{ij} - a_{ij} = 0

, isliye result same order ka zero matrix

O

hai — additive-inverse law action mein.

Kya ek

3 \times 3

matrix ko apne transpose ke saath add kiya ja sakta hai, aur kaun si special structure nazar aati hai?

Haan (

3 \times 3

plus

3 \times 3

);

A + A^{T}

hamesha symmetric hoti hai kyunki slot

(i, j)

aur

(j, i)

dono

a_{ij} + a_{j i}

ke barabar hote hain.

Agar

A

koi bhi

m \times n

matrix hai, to kya

A - O

kabhi

A

se alag hota hai?

Nahi — jab tak

O

m \times n

zero hai,

A - O = A

har slot mein; ek shape mismatch ise "alag" ki jagah undefined kar dega.

Do matrices ke orders

m \times n

aur

n \times m

hain jahan

m \neq = n

; kya

A + B

A - B

A + B^{T}

A - B^{T}

mein se koi bhi compute ho sakta hai?

Sirf

A + B^{T}

aur

A - B^{T}

B

ko transpose karne se woh

m \times n

ho jaata hai

A

se match karne ke liye. Raw

A \pm B

undefined rehte hain kyunki orders alag hain.

Recall Ek-line summary

Same order andar, same order bahar — har slot apne twin ke saath pair karta hai, $+$ saare achhe laws rakhta hai, aur $-$ unhe saare rakhta hai sirf sides swap karne ki freedom chhod kar.

Connections

Matrix operations — addition, subtraction (conditions) — parent topic jise yeh traps drill karte hain.
Matrix Notation and Terminology — order $m \times n$ jo decide karta hai "defined vs undefined".
Zero Matrix and Identity Matrix — kyun $O$ (na ki $I$ ) additive identity hai.
Scalar Multiplication of Matrices — $A + A = 2 A$ aur distributive traps ko power deta hai.
Transpose of a Matrix — $A + B^{T}$ misconception.
Matrix Multiplication — rival dimension rule jise log galti se borrow karte hain.
System of Linear Equations — jahan same-shape matrices combine karne ka meaning hota hai.