Worked examples — Matrix definition — rows, columns, order, elements

2.6.1 · D3 · Maths › Matrices & Determinants — Introduction › Matrix definition — rows, columns, order, elements

Yeh page ek drill hall hai. Parent note ne tumhe chaar ideas sikhaye — rows, columns, order, elements. Yahan hum har tarah ke questions cover karenge jo in ideas se ban sakte hain, ek worked example har scenario ke liye, taaki koi bhi matrix problem tumhare liye naya na rahe.

Shuru karne se pehle, notation ke baare mein ek promise. Jo bhi hum use karte hain, yahan banaya gaya hai:

• Ek matrix numbers ka ek grid hai jo brackets ke andar hota hai.
• Order $m\times n$ ka matlab hai "$m$ rows lambi, $n$ columns chaudi" — pehle rows padhte hain.
• $a_{ij}$ ka matlab hai "row $i$, column $j$ mein baitha number" — phir se row pehle, column doosra ("RC").
• $i$ aur $j$ hamesha 1 se start hote hain, kabhi 0 se nahi.

Agar inme se kuch shaky lagta hai, toh parent note inhe bilkul shuruaat se banata hai. Ab se hum inhe tezi se use karte hain.

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The scenario matrix

Har matrix-reading ya matrix-building question jo tumse poocha ja sakta hai, in cells mein se kisi ek mein aata hai. Neeche ke worked examples mein us cell ka label diya gaya hai jo woh cover karta hai, taaki milke poori table fill ho jaye.

Cell	Scenario	Covered by
A	Ek chhote square grid se ek element padhna	Example 1
B	Rectangular (non-square) grid — order commutative nahi	Example 2
C	Degenerate shapes: ek akela number, ek row vector, ek column vector	Example 3
D	Empty / edge index — ek aisa element maangna jo exist nahi karta	Example 4
E	Formula $a_{ij}=f(i,j)$ se matrix banana signs & ek $\max /\min$ twist ke saath	Example 5
F	Sign-sensitive build: alternating $(-1{)}^{i+j}$ pattern (har sign case)	Example 6
G	Real-world word problem — data ki ek table ko matrix mein badalna aur use padhna	Example 7
H	Exam twist — do matrices equal hain, unknowns ke liye solve karo (equality se link)	Example 8

Related ideas jinhe hum chhoote hain: Matrix Equality, Types of Matrices, Transpose of Matrix.

Upar wali figure dekho: wohi $2\times 3$ grid jisme red arrow pehle neeche jaata hai (woh row $i$ pick karta hai) aur blue arrow across jaata hai (woh column $j$ pick karta hai). Yeh akela picture $a_{ij}$ ki poori grammar hai. Ise apne dimag mein har example ke liye rakhna.

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[!example] Example 1 — Cell A: ek square grid padhna

$P=\left[\begin{array}{ccc}8& -3& 5\\ 0& 4& -6\\ 2& 9& -1\end{array}\right]$

Order nikalo, phir $p_{13}$, $p_{31}$ aur diagonal element $p_{22}$.

Step 1 — Rows count karo, phir columns. 3 horizontal lines, 3 vertical lines → order $3\times 3$. Yeh step kyun? Shape, contents se pehle aati hai; tum ek 3-row matrix mein row 4 nahi maang sakte.

Step 2 — $p_{13}$: row 1 tak neeche jao, column 3 tak across jao. Row 1 hai $[8-35]$; teesri entry $5$ hai. Toh $p_{13}=5$. Yeh step kyun? Red arrow (neeche) row pick karta hai, blue arrow (across) column pick karta hai — bilkul wahi figure ki tarah.

Step 3 — $p_{31}$: row 3 tak neeche jao, column 1 tak across jao. Row 3 hai $[29-1]$; pehli entry $2$ hai. Toh $p_{31}=2$. Yeh step kyun? Subscripts swap karne se tum bilkul alag cell par pahunch jaate ho — $5\ne 2$, toh $p_{13}\ne p_{31}$.

Step 4 — $p_{22}$: row 2, column 2. Row 2 hai $[04-6]$; doosri entry $4$ hai. Toh $p_{22}=4$.

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[!example] Example 2 — Cell B: rectangle, order commutative nahi hai

$Q=\left[\begin{array}{cccc}5& -3& 0& 1\\ 2& 7& -4& 6\end{array}\right]$

Order nikalo, elements ki total sankhya, $q_{14}$, $q_{23}$, aur batao ki $Q$ square hai ya nahi.

Step 1 — Order. 2 rows, 4 columns → order $2\times 4$. Yeh step kyun? $2\times 4$ aur $4\times 2$ alag shapes hain; ordinary multiplication ki tarah jahan $2\times 4=4\times 2$, matrix order commute nahi karta.

Step 2 — Total elements $=m\times n=2\times 4=8$. Yeh step kyun? 2 rows mein se har ek mein 4 numbers hain, aur $2\times 4$ inhe bina overlap ke count karta hai.

Step 3 — $q_{14}$: row 1, column 4. Row 1 hai $[5-301]$; 4th entry $1$ hai. Toh $q_{14}=1$.

Step 4 — $q_{23}$: row 2, column 3. Row 2 hai $[27-46]$; 3rd entry $-4$ hai. Toh $q_{23}=-4$.

Step 5 — Square? Square ke liye rows $=$ columns chahiye. Yahan $2\ne 4$, toh square nahi (yeh rectangular hai).

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[!example] Example 3 — Cell C: degenerate shapes

Sabse chhoti matrices idea ko uski buniyad tak le aati hain. Teen degenerate cases:

(a) Ek akela number as a matrix $S=\left[\begin{array}{c}7\end{array}\right]$. Order $1\times 1$. Sirf ek element $s_{11}=7$. Yeh ek square matrix hai (rows $=$ columns $=1$).

(b) Ek row vector $R=\left[\begin{array}{cccc}3& -1& 5& 0\end{array}\right]$. Order $1\times 4$: ek row, chaar columns. Yahan pehla index 1 par stuck hai, sirf doosra move karta hai: $r_{11}=3,r_{12}=-1,r_{13}=5,r_{14}=0$.

(c) Ek column vector $C=\left[\begin{array}{c}2\\ -7\\ 4\end{array}\right]$. Order $3\times 1$: teen rows, ek column. Ab doosra index 1 par stuck hai: $c_{11}=2,c_{21}=-7,c_{31}=4$.

Yeh kyun important hai? Yeh poore topic ke limiting cases hain: ek row vector "ek aisi matrix hai jisme rows khatam ho gayi", column vector "columns khatam ho gayi", aur $1\times 1$ "dono ek saath khatam". Har general rule yahan bhi kaam karna chahiye, aur karta hai — frozen index bas kabhi change nahi hota.

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[!example] Example 4 — Cell D: woh element jo exist nahi karta

Example 2 ke $Q$ (order $2\times 4$) ko use karke, ek question $q_{34}$ aur $q_{25}$ maangta hai. Inhe dhundho.

Step 1 — $q_{34}$ test karo: row 3. Order $2\times 4$ hai, toh valid rows hain $i\in \{1,2\}$. Row $3$ out of range hai. Yeh step kyun? Index $i$ kabhi $m$ se zyada nahi ho sakta; ek 2-row matrix mein row 3 maangne ka koi jawab nahi — element undefined hai, zero nahi.

Step 2 — $q_{25}$ test karo: column 5. Valid columns hain $j\in \{1,2,3,4\}$. Column $5$, $n=4$ se zyada hai. Yeh step kyun? Doosra index kabhi $n$ se zyada nahi ho sakta. $q_{25}$ undefined hai.

Step 3 — Allowed ranges batao. Kisi bhi $m\times n$ order ke matrix ke liye: $1\le i\le m$ aur $1\le j\le n$. Isse bahar kuch bhi ek non-existent element hai.

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[!example] Example 5 — Cell E: ek twist wale formula se banana

$3\times 3$ matrix $A$ banao jisme $a_{ij}=\max (i,j)-\min (i,j).$ ($\max (i,j)$ ka matlab hai "$i$ aur $j$ mein se bada"; $\min (i,j)$ chhhota. Unka difference bas dono indices kitne door hain yeh hai.)

Step 1 — Diagonal, $i=j$. $\max (i,i)-\min (i,i)=i-i=0$. Toh $a_{11}=a_{22}=a_{33}=0$. Yeh step kyun? Jab row aur column indices same hote hain, gap zero hota hai — ek poora diagonal free mein $0$s ke saath.

Step 2 — Off-diagonal. Value $|i-j|$ hai, dono indices ke beech ki doori:

• $a_{12}=|1-2|=1$, $a_{13}=|1-3|=2$, $a_{23}=|2-3|=1$
• $a_{21}=|2-1|=1$, $a_{31}=|3-1|=2$, $a_{32}=|3-2|=1$

Yeh step kyun? Do numbers ka $\max -\min$ exactly unka absolute difference hota hai, jo negative nahi ho sakta — toh is build mein koi sign trap nahi hai.

Step 3 — Assemble karo. $A=\left[\begin{array}{ccc}0& 1& 2\\ 1& 0& 1\\ 2& 1& 0\end{array}\right]$

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[!example] Example 6 — Cell F: har sign, alternating pattern

$3\times 3$ matrix $B$ banao jisme $b_{ij}=(-1{)}^{i+j}(i+j).$ Yahan $(-1{)}^{i+j}$ ek sign switch hai: yeh $+1$ hota hai jab $i+j$ even ho aur $-1$ jab $i+j$ odd ho. Hume dono sign cases sahi se handle karne hain.

Step 1 — Sign rule padho. $i+j$ even $\Rightarrow$ factor $+1$; $i+j$ odd $\Rightarrow$ factor $-1$. Yeh step kyun? Har cell mein parity sahi karna hi poori mushkil hai; magnitude $(i+j)$ easy hai.

Step 2 — Row 1 ($i=1$):

• $b_{11}$: $i+j=2$ even → $+(2)=2$
• $b_{12}$: $i+j=3$ odd → $-(3)=-3$
• $b_{13}$: $i+j=4$ even → $+(4)=4$

Step 3 — Row 2 ($i=2$):

• $b_{21}$: $3$ odd → $-3$
• $b_{22}$: $4$ even → $4$
• $b_{23}$: $5$ odd → $-5$

Step 4 — Row 3 ($i=3$):

• $b_{31}$: $4$ even → $4$
• $b_{32}$: $5$ odd → $-5$
• $b_{33}$: $6$ even → $6$

Step 5 — Assemble karo. $B=\left[\begin{array}{ccc}2& -3& 4\\ -3& 4& -5\\ 4& -5& 6\end{array}\right]$

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[!example] Example 7 — Cell G: ek real-world word problem

Ek snack stall 3 din ke liye 2 items (samosas, chai) ki bikri record karta hai:

• Monday: 40 samosas, 55 chai
• Tuesday: 30 samosas, 48 chai
• Wednesday: 52 samosas, 60 chai

Ise matrix $D$ ke roop mein likho jisme rows = days aur columns = items hoon. Phir $d_{31}$ aur $d_{12}$ padho, aur batao dono ka kya matlab hai.

Step 1 — Layout fix karo. Rows = days (3 hain), columns = items (2 hain) → order $3\times 2$. Yeh step kyun? Rows aur columns ka matlab chunna modelling step hai; phir har element ka ek definite real-world reading hota hai.

Step 2 — Grid bharo (samosas column 1 mein, chai column 2 mein): $D=\left[\begin{array}{cc}40& 55\\ 30& 48\\ 52& 60\end{array}\right]$

Step 3 — $d_{31}$ padho: row 3 (Wednesday), column 1 (samosas) $=52$. Matlab: Wednesday ko 52 samosas biki.

Step 4 — $d_{12}$ padho: row 1 (Monday), column 2 (chai) $=55$. Matlab: Monday ko 55 chai biki.

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[!example] Example 8 — Cell H: exam twist, unknowns ke liye solve karo

Do matrices equal di gayi hain (yeh equal matrices hain, matlab same order aur har matching entry equal): $\left[\begin{array}{cc}x+y& 3\\ 5& x-y\end{array}\right]=\left[\begin{array}{cc}8& 3\\ 5& 2\end{array}\right].$ $x$ aur $y$ nikalo.

Step 1 — Entry by entry match karo. Equal matrices har corresponding element ko equal force karti hain:

• top-left: $x+y=8$
• top-right: $3=3$ (automatically true — koi info nahi)
• bottom-left: $5=5$ (automatically true)
• bottom-right: $x-y=2$

Yeh step kyun? Matrix equality entry-wise hoti hai; do useful cells ordinary equations ki ek pair dete hain.

Step 2 — System solve karo. Dono equations add karo: $(x+y)+(x-y)=8+2\Rightarrow 2x=10\Rightarrow x=5.$ Yeh step kyun? Add karne se $y$ cancel hota hai, aur $x$ seedha isolate ho jaata hai.

Step 3 — Back-substitute karo. $x+y=8$ se: $5+y=8\Rightarrow y=3$.

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[!mnemonic] Yeh yaad rakhke jao

• RC: Row pehle Column, hamesha. Pehle neeche, phir across (figure mein do arrows dekho).
• Order kabhi commute nahi karta: $m\times n\ne n\times m$ as a shape, chahe count $m\cdot n$ same ho.
• Indices ek box mein rehte hain: $1\le i\le m$, $1\le j\le n$. Box ke bahar → element exist nahi karta.
• Build formulas: pehle blank grid likho, phir har $(i,j)$ feed karo — aur kisi bhi $(-1{)}^{i+j}$ ke liye parity dekho.

[!recall]- Quick self-test