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

2.6.2 · D3 · Maths › Matrices & Determinants — Introduction › Types of matrices — row, column, square, diagonal, identity,

Is page par topic Types of Matrices se aane wale har tarah ke sawaal dhundhe gaye hain. Kisi bhi worked example se pehle, hum ek scenario matrix dete hain: har distinct case ki ek checklist. Phir har worked example us cell ke saath tag hota hai jise woh cover karta hai, taaki end mein koi surprises na bachein.

Yahan sab kuch ek operation par tikaa hai jo tumhe pehle se clearly samajh aana chahiye — transpose, likha jaata hai $A^T$. Iska rule simple hai: $A$ ke row $i$, column $j$ wali entry $A^T$ ke row $j$, column $i$ par chali jaati hai. Symbols mein, $(A^T{)}_{ij}=A_{ji}$. Ise poore grid ko apne main diagonal (top-left se bottom-right wali staircase) ke upar paltne ki tarah socho, jaise usi crease par ek kagaz fold karo. Agar woh fold wali picture abhi bhi fuzzy hai, pehle Matrix Transpose revisit karo.

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The scenario matrix

Matrix types par har exam question actually inhi cells mein se ek hota hai. Hum sab ko cover karenge.

Cell	Case class	Tricky kyon lagta hai	Example
A	Shape se classify karna (row / column / square)	order $m\times n$ vs $n\times m$	Ex 1
B	Diagonal vs identity pahchaanna	diagonal ke liye zeros on diagonal allowed, identity ke liye nahi	Ex 2
C	Degenerate: zero matrix aur $1\times 1$ matrix	kya $[0]$ symmetric hai? skew? dono?	Ex 3
D	Symmetry verify karna mixed signs ke saath	negative aur positive mirror pairs	Ex 4
E	Skew-symmetry verify karna, bura diagonal pakadna	non-zero diagonal ⇒ turant fail	Ex 5
F	Kisi bhi square matrix ko split karna symmetric + skew parts mein	universal decomposition	Ex 6
G	Unknowns solve karke ek type force karna (exam twist)	$a_{ij}=a_{ji}$ se equations banao	Ex 7
H	Word problem — real situation se symmetry	distances / mutual relations symmetric hoti hain	Ex 8
I	Limiting / boundary: symmetric + skew ka sum, aur $A+A^T$ hamesha symmetric	ek general fact prove karo, koi number nahi	Ex 9

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Example 1 — Cell A: shape se classify karna

Forecast: Aage padhne se pehle har ek ka order (rows × columns) aur ek type-name guess karo.

1. $P$ ke rows aur columns count karo. Ek row, teen columns ⇒ order $1\times 3$. Yeh step kyon? Shape sirf counting se decide hoti hai; "row matrix" ki definition hi exactly ek row hai.
2. $P$ ek row matrix hai. Yeh square nahi hai (rows $\ne$ columns), isliye iska koi determinant nahi hai.
3. $Q$ count karo. Teen rows, ek column ⇒ order $3\times 1$ ⇒ column matrix.
4. $R$ count karo. Do rows, do columns ⇒ order $2\times 2$ ⇒ square matrix. Square hona determinants, eigenvalues, aur inverses ka entry ticket hai (dekho Determinants, Inverse of a Matrix).

Verify: Total entries order se match karni chahiye. $P$: $1\times 3=3$ entries ✓. $Q$: $3\times 1=3$ ✓. $R$: $2\times 2=4$ ✓.

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Example 2 — Cell B: diagonal vs identity

Forecast: Identity kaun sa hai? Kya $D_2$ abhi bhi "diagonal" hai phir bhi jab ek diagonal entry $0$ hai?

1. Har ek ke off-diagonal entries ($a_{12},a_{21}$) check karo. Sab $0$ hain, isliye teeno definition ke hisaab se diagonal matrices hain ($a_{ij}=0$ jab bhi $i\ne j$). Yeh step kyon? "Diagonal" sirf off-diagonal ke zero hone ki parwah karta hai — diagonal khud kya rakhta hai, iske baare mein kuch nahi kehta.
2. $D_2$ ka diagonal dekho. Usme ek $0$ hai (top-left). Yeh bilkul allowed hai — ek diagonal matrix apne diagonal par zeros rakh sakta hai. Figure dekho: $D_2$ phir bhi poori tarah staircase par hi rehta hai.
3. Identity test: diagonal aur har diagonal entry $1$ ke barabar ho. Sirf $D_3$ pass karta hai ⇒ $D_3=I_2$.
4. Scalar test: diagonal jisme saari diagonal entries equal hon. $D_3$ (sab $1$) qualify karta hai; $D_1$ (entries $2,5$) aur $D_2$ (entries $0,3$) nahi karte.

Verify: Identity ko $AI=A$ satisfy karna chahiye. $A=\left[\begin{array}{cc}7& 8\\ 9& 1\end{array}\right]$ lo: $A{D}_3=A$ ✓ (VERIFY mein check kiya). Aur $D_1\ne I$ kyunki $2\ne 1$.

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Example 3 — Cell C: degenerate cases

Forecast: Kya ek matrix symmetric aur skew-symmetric dono ek saath ho sakta hai? Step 1 se pehle guess karo.

1. $O$ ka transpose lo. Ek all-zero grid ko flip karne se kuch nahi badalta: $O^T=O$. Toh $O$ symmetric hai. Yeh step kyon? Symmetry bilkul wahi equation hai $A^T=A$; hum ise directly test karte hain.
2. $O$ par skew test: $O^T=-O$ chahiye. Lekin $-O=O$ (zero ko negate karne par zero milta hai), aur $O^T=O$, isliye $O^T=-O$ bhi hold karta hai. Toh $O$ skew-symmetric bhi hai. Yeh step kyon? Zero matrix unique case hai jo dono conditions satisfy karta hai — kyunki $A^T=A$ aur $A^T=-A$ milke $A=-A$ force karte hain, yaani $A=O$.
3. $M=[7]$ ka transpose lo. Ek $1\times 1$ matrix apna khud ka transpose hota hai (flip karne ke liye kuch nahi): $M^T=[7]=M$. Toh $M$ symmetric hai.
4. $M$ par skew test: $M^T=-M$ chahiye, yaani $7=-7$. Galat. Toh $[7]$ skew-symmetric nahi hai. ($1\times 1$ skew ke liye, single entry diagonal par hoti hai, aur skew matrix ke diagonal entries $0$ hone chahiye.)

Verify: Sirf skew $1\times 1$ matrix $[0]$ hai, kyunki $a_{11}=-a_{11}\Rightarrow a_{11}=0$. ✓

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Example 4 — Cell D: mixed signs ke saath symmetry verify karna

Forecast: Symmetry ka matlab hai har entry diagonal ke paar apne mirror ke barabar ho — negatives bhi. Haan/nahi guess karo.

1. Mirror pairs $(a_{ij},a_{ji})$ list karo $i<j$ ke liye.
   • $a_{12}=-2$ aur $a_{21}=-2$: equal ✓ (dono negative — sign match hona chahiye, flip nahi).
   • $a_{13}=4$ aur $a_{31}=4$: equal ✓
   • $a_{23}=-6$ aur $a_{32}=-6$: equal ✓ Yeh step kyon? Symmetry hai $a_{ij}=a_{ji}$; hume sirf diagonal ke upar ki entries ka neeche ke reflections se match karna hai.
2. Diagonal entries hamesha $a_{ii}=a_{ii}$ satisfy karte hain, isliye woh kabhi symmetry nahi rokti.
3. Saare pairs match karte hain ⇒ $A$ symmetric hai, $A^T=A$.

Verify: $A^T$ compute karo aur $A$ se entrywise compare karo (VERIFY mein kiya, equal return hota hai). ✓

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Example 5 — Cell E: bura diagonal pakadna

Forecast: Unme se ek pehli nazar mein hi fail ho jaata hai. Kaun, aur kyon?

1. Pehle $C$ ka diagonal scan karo. $c_{11}=2\ne 0$. Ek skew-symmetric matrix ke saare diagonal entries $0$ hone chahiye (kyunki $a_{ii}=-a_{ii}\Rightarrow a_{ii}=0$). Isliye $C$ instantly reject ho jaata hai — koi transpose ki zaroorat nahi. Yeh step kyon? Zero-diagonal rule sabse fast disqualifier hai; arithmetic karne se pehle ise use karo.
2. $B$ ka diagonal: saare zeros ✓. Gate pass.
3. $B$ ke off-diagonal pairs check karo $a_{ij}=-a_{ji}$ ke liye:
   • $b_{12}=3,b_{21}=-3$: $3=-(-3)$ ✓
   • $b_{13}=-1,b_{31}=1$: $-1=-(1)$ ✓
   • $b_{23}=5,b_{32}=-5$: $5=-(-5)$ ✓
4. Isliye $B$ skew-symmetric hai, $B^T=-B$.

Verify: $B^T=-B$ entrywise check kiya; $C$ fail karta hai kyunki $C^T\ne -C$. ✓

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Example 6 — Cell F: kisi bhi square matrix ko split karna

Forecast: Har square matrix is tarah split hota hai, uniquely. $A$ aur $A^T$ se $S$ aur $K$ ka formula guess karo.

1. Transpose compute karo. $A^T=\left[\begin{array}{cc}4& 6\\ 2& 8\end{array}\right]$. Yeh step kyon? Poora decomposition $A$ aur uske mirror $A^T$ se bana hai.
2. Symmetric part $S=\frac{1}{2}(A+A^T)$. $S=\frac{1}{2}\left[\begin{array}{cc}8& 8\\ 8& 16\end{array}\right]=\left[\begin{array}{cc}4& 4\\ 4& 8\end{array}\right]$ Yeh step kyon? $A+A^T$ hamesha symmetric hota hai (iska transpose $A^T+A$ hai, same cheez), isliye uska aadha symmetric piece hai.
3. Skew-symmetric part $K=\frac{1}{2}(A-A^T)$. $K=\frac{1}{2}\left[\begin{array}{cc}0& -4\\ 4& 0\end{array}\right]=\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right]$ Yeh step kyon? $A-A^T$ hamesha skew hota hai (iska transpose $A^T-A=-(A-A^T)$ hai), jo skew piece deta hai — dhyaan do iska diagonal zero hai, jaise hona chahiye.
4. Reassemble: $S+K=\left[\begin{array}{cc}4& 4\\ 4& 8\end{array}\right]+\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right]=\left[\begin{array}{cc}4& 2\\ 6& 8\end{array}\right]=A$ ✓

Verify: $S^T=S$, $K^T=-K$, aur $S+K=A$ (teeno VERIFY mein). ✓

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Example 7 — Cell G: unknowns solve karna (exam twist)

Forecast: Symmetry ek equation force karti hai. Guess karo yeh kitne unknowns pin down kar sakti hai.

1. Off-diagonal pair ke liye symmetry condition likho: $a_{12}=a_{21}$, yaani $x+2=2y-1.$ Yeh step kyon? Symmetry sirf mirror pairs ko constrain karti hai; diagonal ($3,7$) free hai.
2. Ek equation, do unknowns ⇒ ek line par infinitely many solutions: $x=2y-3$. Yeh step kyon? Hume yeh honestly report karna hai — ek akela symmetry equation dono variables fix nahi kar sakta.
3. Ek sample solution lo: $y=2$ lo $\Rightarrow x=2(2)-3=1$. Tab $a_{12}=1+2=3$ aur $a_{21}=2(2)-1=3$ ✓.

Verify: $x=1,y=2$ ke saath: $A=\left[\begin{array}{cc}3& 3\\ 3& 7\end{array}\right]$, aur $A^T=A$. ✓

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Example 8 — Cell H: word problem (reality se symmetry)

Forecast: X se Y ki distance Y se X ki distance ke barabar hai. Yeh kya force karta hai?

1. Matrix fill karo (towns X, Y, Z order mein): $A=\left[\begin{array}{ccc}0& 12& 8\\ 12& 0& 15\\ 8& 15& 0\end{array}\right]$ Yeh step kyon? Har entry ek physical distance hai; diagonal $0$ hai kyunki ek town khud se zero km door hai.
2. Physics observe karo: distance mutual hoti hai — X→Y = Y→X. Isliye $a_{ij}=a_{ji}$ saare pairs ke liye. Yeh step kyon? Yahi symmetry condition hai, jo naturally ek real property se aati hai.
3. Isliye $A$ symmetric hai. (Yeh skew nahi hai: skew ke liye $a_{ij}=-a_{ji}$ chahiye hoga, yaani negative distances — physically impossible.)

Verify: $A^T=A$ entrywise ✓. Dhyaan do yahan zero diagonal skew ki requirement ke saath ek coincidence hai lekin $A$ skew nahi hai kyunki off-diagonals positive hain, sign-flipped nahi.

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Example 9 — Cell I: ek general fact prove karna (limiting case)

Forecast: Proof ke liye koi numbers nahi chahiye — sirf transpose rules. Guess karo kaun se do rules use karoge.

1. Sums ka transpose rule: $(P+Q{)}^T=P^T+Q^T$. Transpose ka transpose: $(A^T{)}^T=A$ (do baar fold karne par original wapas milta hai). Yeh step kyon? Sirf yahi do facts zaroori machinery hain.
2. Unhe apply karo: $S=A+A^T$ lo. Tab $S^T=(A+A^T{)}^T=A^T+(A^T{)}^T=A^T+A=S.$ Kyunki $S^T=S$, $S$ symmetric hai — har $A$ ke liye, koi exception nahi.
3. Concrete test: $A^T=\left[\begin{array}{cc}1& -3\\ 5& 2\end{array}\right]$, isliye $A+A^T=\left[\begin{array}{cc}2& 2\\ 2& 4\end{array}\right],$ jo clearly symmetric hai.

Verify: Numeric case ke liye $(A+A^T{)}^T=A+A^T$ ✓.

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Recall

Recall Ek matrix symmetric aur skew-symmetric dono kab ho sakta hai?

Sirf zero matrix ::: kyunki $A^T=A$ aur $A^T=-A$ milke $A=-A\Rightarrow A=O$ dete hain.

Recall Skew-symmetric candidate reject karne ka sabse fast tarika?

Diagonal dekho — koi bhi non-zero diagonal entry use instantly disqualify kar deti hai.

Recall

$A$ ko symmetric + skew parts mein split karne ka formula? $A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T)$.

Recall Distance matrix symmetric kyon hai lekin skew kyon nahi?

Distance mutual hoti hai isliye $a_{ij}=a_{ji}$ (symmetric); distances non-negative hoti hain, kabhi sign-flipped nahi (not skew).