Types of matrices — row, column, square, diagonal, identity, zero, symmetric, skew-symmetric: Question bank

2.6.6 · D5 · Maths › Matrices & Determinants — Introduction › Types of matrices — row, column, square, diagonal, identity,

Yeh reasoning gym hai Types of Matrices ke liye. Yahan koi heavy arithmetic nahi hai — har item ek misconception ya boundary case ko pakadta hai. Prompt padho, apna answer zor se bolo, phir reveal karo.

Is page ki vocabulary (yahan se banai hai, kisi aur note se nahi)

Is page ki har cheez kuch girti-chuni ideas par chalti hai, aur har ek ko hum yahan khud banate hain taaki koi bhi doosre note ko yaad karne par depend na kare.

Kisi cell ka address padhna. Matrix ek numbers ka grid hota hai. Ek cell ko point karne ke liye hum ek do-number address likhte hain subscript mein: $a_{ij}$ ka matlab hai "row $i$ , column $j$ mein baitha number." Pehla subscript row ka count hai, doosra column ka count hai. To $a_{23}$ hai row $2$ , column $3$ ki entry. Poori matrix ko aksar $A$ kaha jaata hai, aur uski entries alag-alag $a_{ij}$ hain.

Order (shape). Hum grid ki shape ko $rows \times columns$ ki tarah describe karte hain. Is poore page mein, jab general shape ke liye letters chahiye hote hain, hum $m$ rows ki tadaad ke liye aur $n$ columns ki tadaad ke liye likhte hain, to $m \times n$ order ki matrix mein $m$ rows aur $n$ columns hote hain. Teesra letter $p$ tab aata hai jab do matrices ko product mein chain karte hain.

Neeche diya figure tumhara visual anchor hai — principal diagonal woh cells ki staircase hai jo top-left se bottom-right tak chalti hai, blue mein dikhayi gayi hai.

Figure — Types of matrices — row, column, square, diagonal, identity, zero, symmetric, skew-symmetric

Definition Vocabulary, simple shabdon mein

Row matrix — ek grid jisme exactly ek row ho: order $1 \times n$ , jaise $[2 - 57]$ . Yeh numbers ki ek horizontal line hoti hai.
Column matrix — ek grid jisme exactly ek column ho: order $m \times 1$ , jaise $3 - 1 4$ . Yeh numbers ka ek vertical stack hota hai.
Square matrix — ek grid jahan rows ki tadaad columns ki tadaad ke barabar ho: order $n \times n$ . Sirf inhi ka principal diagonal kone se kone tak pahunchta hai.
Principal diagonal — woh cells $a_{11}, a_{22}, a_{33}, \dots$ jahan row number column number ke barabar ho (upar wali blue staircase). Baaki sab off-diagonal hain (orange cells).
Transpose $A^{T}$ — poore grid ko us blue diagonal ke paar reflect karo. Entry-wise, row $i$ , column $j$ wali cell, row $j$ , column $i$ wali cell se jagah badal leti hai, to $(A^{T})_{ij} = A_{j i}$ . Neeche mirror picture dekho.
$diag (d_{1}, \dots, d_{n})$ — shorthand hai "ek square matrix banao, $d_{1}, d_{2}, \dots, d_{n}$ ko blue diagonal par rakh do, aur har off-diagonal cell mein $0$ bhar do." To $diag (2, 5) = [2005]$ .
$O$ (zero matrix) — woh matrix jis ki har entry $0$ ho; yeh har order ke liye ek shape mein aati hai, jaise $O_{2 \times 2} = [0000]$ .
$I$ (identity matrix) — woh khaas square matrix jisme har diagonal cell par $1$ ho aur baaki sab jagah $0$ : $I_{n} = diag (1, 1, \dots, 1)$ . Misaal ke tor par $I_{2} = [1001]$ . Yeh $A I = I A = A$ follow karta hai.

Transpose woh ek move hai jo symmetric aur skew-symmetric ke peeche hai, to exactly dekhte hain ki woh kya karta hai — dekho kaise $2$ aur $4$ diagonal ke paar reflect hone par jagah badalte hain:

Ab us mirror move ko poori matrix par lagao aur pucho ki reflection kab kisi khaas jagah jaati hai. Agla figure donon special cases ko side by side rakhta hai:

Ab traps hain. Har item sirf upar ki boxed vocabulary se solve ho sakta hai.

True ya false — justify karo

TF1. Har diagonal matrix symmetric hoti hai.

True — off-diagonal entries sab

0

hoti hain, to

a_{ij} = 0 = a_{j i}

automatically, matlab

A^{T} = A

TF2. Har symmetric matrix diagonal hoti hai.

False — symmetry sirf

a_{ij} = a_{j i}

maangti hai; woh mirrored pairs non-zero bhi ho sakte hain, jaise

[1223]

symmetric hai lekin diagonal nahi.

TF3. Identity matrix ek diagonal matrix hai.

True —

I = diag (1, \dots, 1)

mein diagonal ke bahar zeros hain aur upar ones hain; "sab ones" sirf diagonal entries ka ek khaas choice hai.

TF4. Ek zero square matrix symmetric bhi hai aur skew-symmetric bhi.

True —

O^{T} = O

(symmetric) aur

O^{T} = - O

kyunki

- O = O

(skew-symmetric); yahi akela matrix hai jo dono hai.

TF5. Har square matrix ya to symmetric hai ya skew-symmetric.

False — zyaadatar square matrices dono nahi hoti; jaise

[1023]

mein

A^{T} \neq = A

aur

A^{T} \neq = - A

TF6. Ek skew-symmetric matrix ki saari diagonal entries zero honi chahiye.

True — condition

a_{ii} = - a_{ii}

2 a_{ii} = 0

milta hai, to har diagonal entry

0

hai.

TF7. Koi bhi matrix jisme zero diagonal ho woh skew-symmetric hoti hai.

False — zero diagonal zaroori hai lekin kaafi nahi; off-diagonal pairs ke liye bhi

a_{ij} = - a_{j i}

chahiye, jaise

[0520]

fail ho jaata hai.

TF8. Ek

1 \times 1

matrix

[7]

ek saath row matrix, column matrix, aur square matrix teeno hai.

True — usmein ek row hai, ek column hai, aur row/column count equal hai, to teeno labels ek saath lagte hain.

TF9. Ek row matrix symmetric ho sakti hai.

False — symmetry ko

A^{T} = A

chahiye, lekin ek

1 \times n

row matrix (

n > 1

ke saath) ko transpose karne par

n \times 1

column matrix milti hai jo alag shape ki hai, to equality possible nahi jab tak

n = 1

na ho.

TF10. Agar

A^{T} = A

aur

A^{T} = - A

dono hold karein, to

A = O

True — dono se

A = - A

milta hai, to

2 A = O

, is liye

A = O

; zero matrix hi akela aisaa matrix hai jo ek saath symmetric aur skew-symmetric hai.

TF11. Ek diagonal matrix zero matrix ke barabar ho sakti hai.

True —

diag (0, 0, \dots, 0) = O

; "diagonal" koi bhi diagonal values allow karta hai, including sab zeros.

Error dhundho

SE1. "

[2 - 3 3 - 2]

skew-symmetric hai kyunki off-diagonal entries ek doosre ke negatives hain."

Off-diagonal check pass ho jaata hai, lekin diagonal entries

2

aur

- 2

non-zero hain, jo mandatory

a_{ii} = 0

rule ko violate karta hai; yeh skew-symmetric nahi hai.

SE2. "

diag (2, 0, - 5)

diagonal matrix nahi hai kyunki ek diagonal entry

0

hai."

Diagonal par zero bilkul allowed hai; diagonal ka matlab sirf yeh hai ki off-diagonal entries zero honi chahiye, to yeh valid diagonal matrix hai.

SE3. "Kyunki

[1005]

mein diagonal ke bahar zeros hain, yeh identity matrix hai."

Yeh diagonal hai, lekin identity

I

ke liye additionally har diagonal entry

1

honi chahiye; diagonal par

5

hona isko disqualify karta hai.

SE4. "Matrix

[122435]

symmetric hai kyunki

a_{12} = a_{21} = 2

Symmetry sirf square matrices ke liye defined hai; yeh

2 \times 3

hai (2 rows, 3 columns), to

A^{T}

3 \times 2

ki hogi aur kabhi bhi

A

ke barabar nahi ho sakti.

SE5. "

A O = O

to zero matrix se multiply karna hamesha same-sized zero matrix deta hai jaise

A

Product

A O

zero hai, lekin uski shape multiplication rule follow karti hai, to zaruri nahi ki yeh

A

ki shape se match kare —

m \times n

order ki

A

(rows

\times

columns) ko

n \times p

order ki

O

se multiply karne par result

m \times p

hota hai.

SE6. "

A + O = A

kisi bhi zero matrix

O

ke liye kaam karta hai."

Sirf tab jab

O

ki same order ho jaise

A

ki; addition ke liye matching dimensions chahiye, to

O

A

ki exact size ki zero matrix hona chahiye.

SE7. "Ek symmetric matrix kisi bhi relabelling mein symmetric rehti hai, to uska transpose ek alag matrix hai."

Ek symmetric matrix ke liye transpose literally same matrix hota hai (

A^{T} = A

); transpose karne se kuch bhi nahi badalta.

Why questions

WQ1. Kyun ek skew-symmetric matrix ki diagonal koi information nahi rakhti?

Kyunki har diagonal entry

a_{ii} = - a_{ii}

0

hone par majboor hai, to saari "content" strictly off-diagonal mein rehti hai.

WQ2. Kyun identity matrix ko matrices ka "1" kaha jaata hai?

Kyunki

A I = I A = A

kisi bhi compatible

A

ke liye, bilkul waise jaise ek ordinary number ko

1

se multiply karne par woh unchanged rehta hai (upar banaya identity bilkul yahi "do-nothing" multiplier hai — dekho Matrix Operations).

WQ3. Kyun symmetry ke liye square matrix zaroori hai?

Condition

A^{T} = A

mein

A

ko uske transpose se compare kiya jaata hai; woh dono sirf tab same shape ke hote hain jab rows ki tadaad columns ki tadaad ke barabar ho.

WQ4. Kyun diagonal matrices ko powers tak uthana itna aasaan hai?

Diagonal matrices ko multiply karne par sirf matching diagonal entries multiply hoti hain, koi cross-terms nahi, to

D^{k} = diag (d_{1}^{k}, \dots, d_{n}^{k})

— ek key reason kyun Diagonalization itna prized hai.

WQ5. Kyun sirf square matrices mein determinant aur eigenvalues hote hain?

Yeh notions describe karte hain ki matrix ek space ko usi par kaise map karta hai (same input aur output dimension), jiske liye equal rows aur columns chahiye.

WQ6. Kyun "

a_{ij} = a_{j i}

" guarantee karne ke liye kaafi hai ki

A^{T} = A

Kyunki

(A^{T})_{ij} = A_{j i}

definition se hai (upar wala mirror reflection), to agar

A_{j i} = A_{ij}

har pair ke liye ho, to

A^{T}

aur

A

entry-for-entry match karte hain.

WQ7. Kyun ek diagonal matrix geometrically ek "pure scaling" transformation hai?

Har coordinate axis ko uski diagonal entry se stretch kiya jaata hai, axes ke beech koi mixing nahi, to kuch bhi rotate ya shear nahi hota (dekho Linear Transformations).

WQ8. Kyun ek diagonal matrix ka scaling inverse fail ho sakta hai?

Agar koi bhi diagonal entry

0

hai, to woh axis collapse ho jaata hai aur use undo nahi kiya ja sakta, to matrix ka koi inverse nahi hota.

Edge cases

EC1. Kya yeh khali dikhne wali

2 \times 2

zero matrix

[0000]

diagonal hai?

Haan — sab off-diagonal entries zero hain, diagonal condition satisfy karti hai; uski diagonal entries sirf zero hoti hain.

EC2. Kya koi matrix ek saath diagonal matrix aur skew-symmetric ho sakti hai?

Sirf zero matrix — diagonal non-zero diagonal entries allow karta hai, lekin skew-symmetry unhe sab

0

hone par majboor karti hai, aur off-diagonal entries already

0

hain.

EC3. Ek

1 \times 1

matrix ka transpose kya hai, aur yeh use kya banata hai?

1 \times 1

matrix apne transpose ke barabar hoti hai (flip karne ke liye kuch nahi), to yeh automatically symmetric hai — aur, ek single number hone ke naate, trivially diagonal bhi.

EC4. Kya ek number ko matrix ki tarah treat karne par woh skew-symmetric hoti hai?

1 \times 1

matrix skew-symmetric tab hi hogi jab uski akeli diagonal entry

0

ho, yaani sirf

[0]

qualify karta hai.

EC5. Kya ek column matrix kabhi symmetric ho sakti hai?

Sirf agar woh

1 \times 1

ho; ek se zyaada rows hone par uska transpose ek alag shape ki row matrix hogi, to

A^{T} \neq = A

EC6. Agar ek square matrix ki har entry same constant

c

ke barabar ho, to kab woh symmetric hai?

Hamesha —

a_{ij} = c = a_{j i}

sab pairs ke liye hold karta hai, to ek constant square matrix kisi bhi

c

ke liye symmetric hai (haalaanki skew-symmetric sirf tab jab

c = 0

ho).

EC7. Kya ek diagonal matrix jisme ek repeated value ho jaise

diag (3, 3, 3)

ka koi special name hai?

Haan — yeh ek scalar matrix hai,

3 I

ke barabar; yeh har axis ko identically scale karta hai aur multiplication mein number

3

ki tarah behave karta hai.

Recall Ek-line self-test

Ek aisi matrix batao jo symmetric, diagonal, aur inverse wali ho. ::: Koi bhi $diag (d_{1}, \dots, d_{n})$ jisme sab $d_{i} \neq = 0$ hon — jaise identity $I$ ; yeh diagonal hai (is liye symmetric bhi) aur invertible hai kyunki koi axis collapse nahi hota.