Worked examples — Matrix operations — addition, subtraction (conditions)

2.6.3 · D3 · Maths › Matrices & Determinants — Introduction › Matrix operations — addition, subtraction (conditions)

Yeh page matrix addition aur subtraction ka drill hall hai. Parent note ne rule banaya tha: do matrices element-by-element combine hote hain, aur sirf tab jab unka same order ho (same number of rows aur same number of columns). Yahan hum us rule pe har tarah ki situation daalte hain taaki exam mein koi surprise na mile.

Shuru karne se pehle, ek reminder zero se build karte hain: ek matrix jaise $A=[a_{ij}{]}_{m\times n}$ sirf numbers ka ek rectangle hai. Chhota label $a_{ij}$ ka matlab hai "row $i$, column $j$ mein baitha number". Toh $a_{23}$ matlab 2nd row, 3rd column ka number. Jab hum "corresponding entries add karo" kehte hain, matlab: $A$ ke slot (2,3) ka number plus $B$ ke slot (2,3) ka number, answer ke slot (2,3) mein jaata hai. Agar yeh indexing shaky lagti hai toh Matrix Notation and Terminology dekho.

The scenario matrix

Is topic ka har problem inhi cells mein se ek mein aata hai. Page ka baaki hissa har ek ke liye ek example karta hai.

C1 · Mixed signs, valid order

Forecast: answer cover karo aur top-left entry guess karo. (Yeh hai $3+(-1)$.)

1. Order check karo. Yeh step kyun? Addition sirf identical orders ke liye defined hai — hamesha sabse pehle yahi test karo. Dono $2\times 2$ hain. ✓ Defined.
2. Slot by slot add karo. Yeh step kyun? Definition ke hisaab se $c_{ij}=a_{ij}+b_{ij}$, toh har answer entry sirf same slot ko dekhti hai dono matrices mein. $A+B=\left[\begin{array}{cc}3+(-1)& -5+4\\ -2+6& 0+(-7)\end{array}\right]=\left[\begin{array}{cc}2& -1\\ 4& -7\end{array}\right]$
3. Sign rules dhyan se dekho. Yeh step kyun? Negative add karna matlab subtract karna; $-5+4=-1$, $0+(-7)=-7$. Sabse common slip yahan hoti hai, layout mein nahi.

Verify: Doosre order mein add karo $B+A$: $-1+3=2$, $4-5=-1$, $6-2=4$, $-7+0=-7$ — bilkul same, jaisa commutativity ka promise hai.

C2 · Invalid order — answer ek word hai

Forecast: dono mein teen numbers hain — tempting lagta hai, hai na?

1. Orders padho. Yeh step kyun? Elements kabhi mat gino; shape gino. $M$ hai $1\times 3$ (ek row, teen columns). $N$ hai $3\times 1$ (teen rows, ek column).
2. Compare karo. Yeh step kyun? Addition ke liye $m$ aur $n$ dono match hone chahiye. Yahan $1\times 3\ne 3\times 1$. ✗
3. Result batao. $M+N$ undefined hai. $N$ mein koi slot $(1,3)$ nahi hai (uska sirf ek column hai), aur $M$ mein koi slot $(3,1)$ nahi hai (uski sirf ek row hai).

Verify: Total element count match karta hai ($3=3$) phir bhi operation fail hoti hai — proof ki "same count" ek trap hai. Yeh exactly parent note ki Mistake 1 hai.

C3 · Zero matrix (additive identity)

Forecast: padhne se pehle guess karo.

1. Order check. Yeh step kyun? $O$ ka same order $2\times 2$ hona chahiye identity banane ke liye — zero matrix har size mein aati hai (dekho Zero Matrix and Identity Matrix).
2. Add karo. Yeh step kyun? $a_{ij}+0=a_{ij}$ har real number ke liye, toh kuch nahi badalta. $A+O=\left[\begin{array}{cc}7+0& -4+0\\ 2+0& 9+0\end{array}\right]=\left[\begin{array}{cc}7& -4\\ 2& 9\end{array}\right]=A$

Verify: Result entry-for-entry $A$ ke barabar hai. $O$ additive identity hai, bilkul waise jaise ordinary numbers mein $0$ behave karta hai.

C4 · Additive inverse

Forecast: sum kya hona chahiye?

1. Har entry negate karo. Yeh step kyun? $-A=[-a_{ij}]$ — har slot ka sign flip karo (yeh $-1$ se scalar multiplication hai, dekho Scalar Multiplication of Matrices). $-A=\left[\begin{array}{cc}-5& 3\\ 0& -8\end{array}\right]$
2. Add karo. Yeh step kyun? $a_{ij}+(-a_{ij})=0$ har slot ke liye, toh hum zero matrix pe aana chahiye. $A+(-A)=\left[\begin{array}{cc}5-5& -3+3\\ 0+0& 8-8\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=O$

Verify: Har entry $0$ hai. Isliye $A-B=A+(-B)$: subtraction sirf inverse add karna hai.

C5 · Degenerate shapes (thin matrices)

Forecast: kya rule skinny matrices ke liye badal jaata hai? (Nahi badalta.)

1. (a) Do $1\times 3$ rows. Yeh step kyun? Same order → element-wise. $\left[\begin{array}{ccc}2+3& 5-5& -1+4\end{array}\right]=\left[\begin{array}{ccc}5& 0& 3\end{array}\right]$.
2. (b) Do $2\times 1$ columns. Yeh step kyun? Har slot subtract karo. $\left[\begin{array}{c}4-1\\ -2-(-6)\end{array}\right]=\left[\begin{array}{c}3\\ 4\end{array}\right]$.
3. (c) Do $1\times 1$ matrices. Yeh step kyun? Ek $1\times 1$ matrix ek number hai jo box pehne hua hai — addition ordinary arithmetic ban jaati hai: $9+(-4)=5$, yaani $\left[\begin{array}{c}5\end{array}\right]$.

Verify: (a) middle entry $5-5=0$; (b) bottom $-2+6=4$; (c) $9-4=5$. Sab consistent — rule vectors aur scalars tak cleanly scale down hota hai.

C6 · Subtraction ki order sensitivity

Forecast: kya ye equal hain? Opposite? Unrelated?

1. $A-B$ compute karo. Yeh step kyun? $d_{ij}=a_{ij}-b_{ij}$. $A-B=\left[\begin{array}{cc}8-5& 1-(-4)\\ -3-7& 2-0\end{array}\right]=\left[\begin{array}{cc}3& 5\\ -10& 2\end{array}\right]$
2. $B-A$ compute karo. Yeh step kyun? Roles swap karo; har entry sign flip karti hai kyunki $b_{ij}-a_{ij}=-(a_{ij}-b_{ij})$. $B-A=\left[\begin{array}{cc}5-8& -4-1\\ 7-(-3)& 0-2\end{array}\right]=\left[\begin{array}{cc}-3& -5\\ 10& -2\end{array}\right]$
3. Compare karo. Yeh step kyun? Notice karo $B-A=-(A-B)$: har sign opposite hai. Subtraction anti-commutative hai, addition ki tarah nahi.

Verify: Dono results add karo: $(A-B)+(B-A)$ ko $O$ dena chahiye. $3+(-3)=0$, $5+(-5)=0$, $-10+10=0$, $2+(-2)=0$. ✓

C7 · Transpose trap (do figures)

Forecast: konsa kaam karta hai, aur kya transposing problem "fix" karti hai ya question badal deti hai?

Neeche ki picture dikhati hai kyun $A+B$ fail hota hai: grids overlap nahi karti.

1. $A+B$ test karo. Yeh step kyun? $2\times 3$ vs $3\times 2$ — mismatch → undefined (cell C2 phir se).
2. $B$ transpose karo. Yeh step kyun? Transpose rows aur columns swap karta hai: $b_{ij}^{\mathsf{T}}=b_{ji}$ (dekho Transpose of a Matrix). Toh $B^{\mathsf{T}}$ ban jaata hai $2\times 3$ aur sizes ab match karti hain. $B^{\mathsf{T}}=\left[\begin{array}{ccc}10& 20& 30\\ 40& 50& 60\end{array}\right]$
3. Add karo. Yeh step kyun? Ab defined hai: $A+B^{\mathsf{T}}=\left[\begin{array}{ccc}1+10& 2+20& 3+30\\ 4+40& 5+50& 6+60\end{array}\right]=\left[\begin{array}{ccc}11& 22& 33\\ 44& 55& 66\end{array}\right]$

Agla figure dikhata hai ki transpose $B$ ko reshape karke uske slots $A$ ke saath align karta hai.

Verify: Entry $(2,3)$: $6+b_{23}^{\mathsf{T}}=6+b_{32}=6+60=66$. ✓

C8 · Unknown matrix ke liye solve karo

Forecast: "$+$ ek matrix" ko kaunsa tool undo karta hai?

1. Dono sides se known matrix subtract karo. Yeh step kyun? Numbers ki tarah hi: $X$ isolate karne ke liye, constant matrix ka additive inverse add karo. Har entry ordinary algebra maanti hai. $X=\left[\begin{array}{cc}5& 4\\ -2& 3\end{array}\right]-\left[\begin{array}{cc}2& -1\\ 0& 3\end{array}\right]$
2. Element-wise subtraction. Yeh step kyun? Valid — dono $2\times 2$ hain. $X=\left[\begin{array}{cc}5-2& 4-(-1)\\ -2-0& 3-3\end{array}\right]=\left[\begin{array}{cc}3& 5\\ -2& 0\end{array}\right]$

Verify: Plug back karo: $\left[\begin{array}{cc}3& 5\\ -2& 0\end{array}\right]+\left[\begin{array}{cc}2& -1\\ 0& 3\end{array}\right]=\left[\begin{array}{cc}5& 4\\ -2& 3\end{array}\right]$. ✓ Right side se match karta hai.

C9 · Real-world word problem

Forecast: konsa operation "total" hai, konsa "difference"?

1. Confirm karo ki structure match karti hai. Yeh step kyun? Dono $2\times 2$ shop×product hain aur har slot ka identical matlab hai — parent note jo semantic alignment demand karta hai. ✓
2. Total = $J+F$. Yeh step kyun? Do months mein same relationship combine karna → add karo. $J+F=\left[\begin{array}{cc}52& 27\\ 38& 44\end{array}\right]$
3. Difference = $F-J$. Yeh step kyun? "Feb mein kitna zyada" = February minus January, entry by entry. $F-J=\left[\begin{array}{cc}-8& 3\\ 2& -6\end{array}\right]$ Padho: Shop 1 Product 1 ne February mein 8 kam becha (negative = drop), Shop 1 Product 2 ne 3 zyada becha, aur aage bhi aisa hi.

Verify: Total ka top-left $30+22=52$ ✓. Difference ka bottom-right $19-25=-6$ ✓, aur $(J+F)$ ne same slots sum kiye jo $(F-J)$ ne difference kiye — consistent structure.

C10 · Exam twist — scalar-weighted combination

Forecast: pehle scale karo ya add karo? (Pehle scale — yeh har entry pe distribute hota hai.)

1. Har matrix scale karo. Yeh step kyun? $kA$ har entry ko $k$ se multiply karta hai (Scalar Multiplication of Matrices); combine karne se pehle yeh hona chahiye. $2A=\left[\begin{array}{cc}4& -2\\ 8& 0\end{array}\right],3B=\left[\begin{array}{cc}3& 9\\ -6& 15\end{array}\right]$
2. Element-wise subtract karo. Yeh step kyun? Dono scaled matrices abhi bhi $2\times 2$ hain → subtraction defined. $2A-3B=\left[\begin{array}{cc}4-3& -2-9\\ 8-(-6)& 0-15\end{array}\right]=\left[\begin{array}{cc}1& -11\\ 14& -15\end{array}\right]$

Verify: Slot $(2,1)$: $2(4)-3(-2)=8+6=14$. ✓ Scalars aur subtraction ek saath karne se same number milta hai — yeh distributive law hai $2A-3B=[2a_{ij}-3b_{ij}]$.

[!recall]- Quick self-test

Connections

• Matrix operations — addition, subtraction (conditions) — parent rule jise yeh examples drill karte hain.
• Matrix Notation and Terminology — $a_{ij}$ slot indexing jo poore note mein use hoti hai.
• Zero Matrix and Identity Matrix — cell C3 ka additive identity $O$.
• Scalar Multiplication of Matrices — cells C4 aur C10.
• Transpose of a Matrix — cell C7 ka reshaping trap.
• Matrix Multiplication — contrast: iske liye alag size condition chahiye.
• System of Linear Equations — cell C9 ki word-problem style.