Matrices — review, operations, types: Question bank

4.5.6 · D5 · Maths › Linear Algebra (Full) › Matrices — review, operations, types

Vocabulary ka reminder jis par ye questions tike hain: ek matrix ek grid of numbers hai jo ek linear transformation ko represent karta hai; product $A B$ ka matlab hai "pehle $B$ karo, phir $A$ "; transpose $A^{⊤}$ rows aur columns ko swap karta hai; singular ka matlab hai $det A = 0$ (koi inverse nahi); orthogonal ka matlab hai $A^{⊤} A = I$ . Neeche har answer "why" re-explain karta hai, taki tumhe kabhi bhi ye cheezein sirf memorize na karni padein.

True or false — justify

Matrix addition commutative hoti hai:

A + B = B + A

same order ki matrices ke liye.

True — addition entrywise hoti hai, aur har number-pair

a_{ij} + b_{ij}

ordinary number commutativity follow karta hai. "Do B first" wala ordering issue sirf multiplication mein aata hai.

Matrix multiplication commutative hoti hai.

False —

A B

ka matlab hai "pehle

B

phir

A

B A

karne se actions ka order badal jaata hai (jaise socks-then-shoes vs shoes-then-socks), isliye generally

A B \neq = B A

Agar

A B

exist karta hai, toh

B A

bhi exist karna chahiye.

False —

A B

ke liye

A

ke columns =

B

ke rows chahiye. Ek

2 \times 3

times

3 \times 2

deta hai

2 \times 2

, lekin reverse hai

3 \times 2

times

2 \times 3 = 3 \times 3

: exist karta hai lekin alag shape ka hai, toh dono equal ho hi nahi sakte.

Har square matrix identity ke saath commute karta hai:

A I = I A = A

True —

I

ek "kuch nahi karo" wala transformation hai, toh

A

se pehle ya baad mein lagane se kuch nahi badalta. Identity hi woh ek matrix hai jo hamesha sabke saath commute karti hai.

Agar

A^{⊤} = A

aur

B^{⊤} = B

, toh

A B

symmetric hai.

False —

(A B)^{⊤} = B^{⊤} A^{⊤} = B A

, jo

A B

ke barabar tab hi hoga jab

A

aur

B

commute karein. Do symmetric matrices ko commute karna zaruri nahi, toh unka product usually symmetric nahi hota.

Kisi bhi skew-symmetric matrix ka diagonal poora zeros ka hota hai.

True — skew ka matlab hai

a_{ij} = - a_{j i}

, toh diagonal par

a_{ii} = - a_{ii}

, isse

2 a_{ii} = 0

force hota hai, yani

a_{ii} = 0

Har orthogonal matrix invertible hoti hai.

True —

A^{⊤} A = I

ka matlab hai

A^{⊤}

hi inverse hai. Orthogonal matrices rotations/reflections hoti hain; ye lengths preserve karti hain aur inhe hamesha undo kiya ja sakta hai.

Agar

det A = 0

toh

A

ka koi inverse nahi hai.

True — determinant measure karta hai ki transformation area/volume ko kitna scale karta hai; zero hona matlab space lower dimension par collapse ho gayi, aur tum usse un-flatten nahi kar sakte. Dekho Determinant and Inverse of a Matrix.

λ (A B) = (λ A) B = A (λ B)

kisi scalar

λ

ke liye.

True — ek scalar poore transformation ke output ko uniformly scale karta hai, toh wo freely kisi bhi factor ki taraf ja sakta hai. Scalars mein koi "order" nahi hota, matrices ki tarah.

Ek sum ka transpose sum of transposes hota hai:

(A + B)^{⊤} = A^{⊤} + B^{⊤}

True — transpose sirf

(i, j) \to (j, i)

entry by entry relabel karta hai, aur addition entrywise hai, toh dono operations ek doosre mein interfere nahi karte.

Spot the error

(A B)^{⊤} = A^{⊤} B^{⊤}

, kyunki transpose products par distribute hota hai."

Galat — sahi reversal law hai

(A B)^{⊤} = B^{⊤} A^{⊤}

. Transpose roles aur order dono reverse karta hai (socks-and-shoes);

A^{⊤} B^{⊤}

ke inner dimensions usually mismatched hote hain aur exist bhi nahi karta.

A B = 0

, toh ya toh

A = 0

hai ya

B = 0

Galat — matrices mein zero divisors hote hain. Jaise

[1000] [0001] = 0

jahan koi bhi factor zero nahi hai. Cancellation sirf invertible matrices ke liye hoti hai.

A B = A C

implies

B = C

Galat — tum

A

tabhi cancel kar sakte ho jab

A

invertible ho. Agar

A

singular hai toh wo alag inputs ko same output par collapse kar sakta hai, toh

B \neq = C

possible hai.

"Matrices multiply karna matlab corresponding entries multiply karna hai, bilkul add karne ki tarah."

Galat — woh Hadamard product hai, jo transformations ko compose nahi karta. Asli multiplication hai (row of

A

)·(column of

B

) as a dot product, jo is wajah se forced hai:

(A B) x = A (B x)

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

matrices ke liye."

Galat — expand karne par milta hai

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

, aur generally

A B \neq = B A

, toh beech waali terms

2 A B

mein combine nahi ho saktin. Binomial shortcut ke liye commutativity chahiye.

"Kyunki

A^{2} = I

, matrix

A

identity honi chahiye."

Galat — reflections bhi

I

tak square hoti hain (do baar reflect = original). Jaise

[10 0 - 1]

square karke

I

deta hai lekin

I

hai nahi. Dekho Eigenvalues and Eigenvectors (eigenvalues

\pm 1

"Ek

2 \times 3

matrix ko ek

3 \times 2

matrix ke saath add kiya ja sakta hai unhe line up karke."

Galat — addition entry

a_{ij}

b_{ij}

ke saath pair karti hai, jo tab hi exist karta hai jab dono dimensions match karein. Alag shapes ka koi natural pairing nahi hota.

A^{⊤} A

aur

A A^{⊤}

same matrix hain."

Galat — non-square

A

ke liye inke sizes bhi alag hote hain (

n \times n

m \times m

), aur square

A

ke liye bhi generally alag hote hain jab tak

A

normal na ho. Yahan bhi order matter karta hai.

Why questions

A B

ke exist hone ke liye inner dimensions match kyun hone chahiye?

Kyunki entry

(A B)_{ij}

ek dot product hai

A

ki ek row aur

B

ke ek column ka; dot product ke liye dono sides par same number of terms chahiye, yani

A

ke columns =

B

ke rows.

Matrix multiplication row·column se kyun define hoti hai, entrywise se kyun nahi?

Taaki

(A B) x = A (B x)

— product ko transformation

B

phir

A

karne ko represent karna chahiye. Entrywise multiplication maps compose nahi karti aur na hi linear systems solve karti.

Transpose product ka order reverse kyun karta hai?

Transpose rows aur columns swap karta hai, effectively map ko "output side se" dekhta hai. Composition ulti taraf se padh ne par dono roles aur order flip ho jaate hain, deta hai

(A B)^{⊤} = B^{⊤} A^{⊤}

Har square matrix uniquely ek symmetric plus skew-symmetric part mein kyun split hoti hai?

Likho

A = \frac{1}{2} (A + A^{⊤}) + \frac{1}{2} (A - A^{⊤})

: pehla piece apne transpose ke barabar hai (symmetric), doosra apne transpose ke negative ke barabar (skew), aur split forced hai.

det A = 0

koi inverse nahi hone ka signal kyun deta hai?

Determinant transformation ka area/volume scaling factor hai. Zero matlab space flat hो gayi, toh alag inputs merge ho jaate hain — woh collapse reverse nahi ho sakta, isliye koi inverse nahi.

Orthogonal matrices lengths kyun preserve karti hain?

Agar

A^{⊤} A = I

, toh

∣ A x ∣^{2} = (A x)^{⊤} (A x) = x^{⊤} A^{⊤} A x = x^{⊤} x = ∣ x ∣^{2}

. Length unchanged rehti hai, toh ye rotation ya reflection hai.

Identity matrix matrices ka multiplicative "1" kyun hai?

Ye "kuch nahi karo" wala transformation hai:

I A = A I = A

. Jaise kisi number ko 1 se multiply karne par wo unchanged rehta hai,

I

ke saath compose karne par koi bhi map unchanged rehta hai.

A B

non-commutative kyun ho sakta hai jab bhi dono diagonal hon?

Hota nahi hai — do diagonal matrices commute karte hain, kyunki har ek sirf independent axes ko scale karta hai aur scaling factors order-freely multiply hote hain. Non-commutativity ke liye axes ka mixing chahiye (rotation, shear).

Edge cases

1 \times n

row matrix aur ek

n \times 1

column matrix ka product kya hota hai?

1 \times 1

matrix — ek single number, bilkul dono vectors ka dot product. Isi liye matrix product entries dot products hoti hain.

Kya

1 \times 1

matrix

[c]

symmetric hai, skew-symmetric hai, ya dono?

Hamesha symmetric (

[c]^{⊤} = [c]

); skew-symmetric sirf tab jab

c = 0

, kyunki skew diagonal entry ko vanish karne par force karta hai. Zero

1 \times 1

matrix dono hai.

Kya koi matrix symmetric aur skew-symmetric dono ho sakti hai?

Haan, lekin sirf zero matrix.

A = A^{⊤}

aur

A = - A^{⊤}

dene par

A = - A

, toh

2 A = 0

, yani

A = 0

Kya zero matrix orthogonal hai?

Nahi — orthogonality require karti hai

A^{⊤} A = I

, lekin zero matrix ke liye

A^{⊤} A = 0 \neq = I

. Zero map sab kuch collapse kar deta hai aur koi lengths preserve nahi karta.

Kisi bhi matrix ko compatible size ki zero matrix se multiply karne par kya hota hai?

Zero matrix milti hai — har output entry ek all-zero row ya column ke saath dot product hai. Ye tab bhi hota hai jab zero product ke liye dono factors ka zero hona zaruri nahi tha.

Kya

1 \times 1

matrix ki multiplication commutative hoti hai?

Haan — ye sirf ordinary number multiplication hai,

[a] [b] = [ab] = [b] [a]

. Non-commutativity genuinely higher-dimensional phenomenon hai.

Kya har non-zero matrix ka inverse hota hai?

Nahi — sirf square, non-singular matrices ka hota hai. Non-square matrices aur

det = 0

wali square matrices (jaise koi bhi matrix jisme zero row ho) ka koi two-sided inverse nahi hota.

Agar

A

3 \times 3

matrix hai jisme do identical rows hain, toh

det A

kya hai, aur kya

A

invertible hai?

det A = 0

, toh

A

singular hai aur non-invertible hai — identical rows ka matlab hai transformation ek dimension collapse kar deta hai, volume zero ho jaata hai.

Recall In traps ki ek-line summary

Entrywise add/scale ke liye, dot-products multiply ke liye; order matter karta hai ( $A B \neq = B A$ , $(A B)^{⊤} = B^{⊤} A^{⊤}$ ); sirf invertible hone par cancel karo; $det = 0$ space collapse kar deta hai; zero products zero factors force nahi karte.

Connections

Matrices — review, operations, types — parent topic jise ye traps stress-test karte hain.
Determinant and Inverse of a Matrix — singularity aur cancellation.
Linear Transformations — non-commutativity ordered actions ke baare mein kyun hai.
Systems of Linear Equations — entrywise multiply kyun fail hoti.
Eigenvalues and Eigenvectors — reflections $I$ tak square hoti hain.
Dot Product and Vectors — har product entry ek dot product hai.