2.6.10 · D1Inverse of 2×2 matrix
Ek 2 × 2 machine hai jo flat space ko reshape karti hai , aur uska inverse woh machine hai jo har point ko exactly wahin wapas rakh deti hai jahan se woh shuru hua tha. Inverse formula samajhne ke liye sirf teen pictures chahiye: ek matrix jo arrows par act karti hai, ek determinant jo usse bani area measure karta hai, aur identity matrix jo "kuch-nahi-karne-wali" machine hai — parent page ki baaki sab cheezein in teeno ke aas-paas ka bookkeeping hai.
Yeh page parent note mein use hone wale har ek symbol ko blank page se build karta hai. Agar wahan koi symbol aata hai, toh usse pehle yahan define kiya gaya hai, ek picture se anchor kiya gaya hai, aur justify kiya gaya hai.
2 × 2 matrix numbers ka ek rectangular grid hota hai. Ek 2 × 2 do rows aur do columns hote hain — char numbers ek box mein:
A = ( a c b d ) A x
Letters a , b , c , d placeholders hain. Unki positions ke naam hain jo tumhe kabhi nahi bhuulne chahiye:
a d main diagonal hain (corner-to-corner line jo ↘ direction mein jaati hai).b c off-diagonal hain (doosre corners).
Intuition Numbers ko box mein arrange kyon karte hain?Matrix sirf storage nahi hai — yeh arrows ko transform karne ki ek recipe hai. Box layout exactly wahi hai jo "row meets column" multiplication rule (Section 4) ko kaam karne deti hai. Figure dekho: main diagonal (orange) aur off-diagonal (teal) ko parent page ke har rule mein alag-alag treat kiya jaata hai, isliye inhe alag dekhna seekhna step one hai.
Ek vector numbers ki ek vertically stacked list hoti hai. 2D mein iske paas do numbers hote hain:
x = ( x 1 x 2 ) x
Picture yeh hai: ek vector origin ( 0 , 0 ) ( x 1 , x 2 ) arrow hota hai. Pehla number batata hai kitna right, doosra kitna upar.
Intuition Is topic ko vectors ki zaroorat kyon hai?Parent note ka pehla sentence kehta hai "a matrix transforms vectors: x ↦ A x vector hota hai , woh sentence meaningless hai. Vectors matrix machine ke inputs aur outputs hain. Dekho Linear transformations and area poori kahani ke liye ki boxes arrows ko kaise move karte hain.
Symbol ↦ x ↦ A x "arrow x A x
2D mein har vector do basic arrows se bana hota hai:
e ^ 1 = ( 1 0 ) ( one step right ) , e ^ 2 = ( 0 1 ) ( one step up ) .
Intuition Yeh dono matrices ke liye kyun matter karte hainYeh raha secret decoder ring: matrix ke columns tumhe batate hain ki yeh dono arrows kahan land karte hain. Pehla column ( c a ) e ^ 1 ( d b ) e ^ 2 A hi transformation ke fingerprints hain.
Definition Matrix × vectorMatrix ko vector se multiply karne ke liye, matrix ki har row vector se milti hai: matching entries ko multiply karo, phir add karo.
( a c b d ) ( x 1 x 2 ) = ( a x 1 + b x 2 c x 1 + d x 2 )
Definition Matrix × matrixSame idea, column by column karo. Answer ka har column (left matrix) hai jo right matrix ke matching column par apply hota hai:
( a c b d ) ( p r q s ) = ( a p + b r c p + d r a q + b s c q + d s )
Intuition "Row meets column" kyon?HUMNE KYA KIYA: pehle box ki har row ko doosre ki har column ke saath pair kiya, across multiply kiya, sum kiya. KYON: yahi ek rule hai jo "machine B A A B hai hi transformations ka composition. Yahi exact expansion parent note ke chaar equations a p + b r = 1 Matrix multiplication .
Worked example Warm-up multiply( 2 3 1 4 ) ( 4 − 3 − 1 2 ) 2 ⋅ 4 + 1 ⋅ ( − 3 ) = 8 − 3 = 5 2 ⋅ ( − 1 ) + 1 ⋅ 2 = 0 3 ⋅ 4 + 4 ⋅ ( − 3 ) = 0 3 ⋅ ( − 1 ) + 4 ⋅ 2 = 5 ( 5 0 0 5 ) 5
Definition Identity matrixI I = ( 1 0 0 1 ) I x = x
Picture: I e ^ 1 ( 0 1 ) e ^ 2 ( 1 0 )
I A A − 1 = I target hai. Dekho Identity matrix .
Definition Determinant of a2 × 2 det A = a d − b c
Par yeh hai kya ? Picture: A parallelogram banaate hain. Determinant us parallelogram ka signed area hai.
Intuition Har sign, aur zero case
Positive det columns apna orientation rakhte hain (column 1 se column 2 ki taraf turn counter-clockwise hai). Area genuine hai.Negative det machine plane ko palat deti hai (jaise mirror); "signed" area negative hai, par uska size phir bhi parallelogram ka area hai.Zero det dono column arrows ek hi line mein point karte hain — parallelogram flat squash ho jaata hai, area = 0
TOPIC KO YAHI KYON CHAHIYE: inverse formula det A Determinant of a 2×2 matrix aur Linear transformations and area .
Common mistake "Determinantab − c d Kyon sahi lagta hai: tum top pair aur bottom pair uthaa lete ho.
Kyon galat hai: rule diagonals pair karta hai, rows nahi. Yeh main diagonal a d b c Fix: det = a d − b c
A − 1 A − 1 A A − 1 = A − 1 A = I − 1 5 − 1 = 5 1 5 ⋅ 5 − 1 = 1 1 I
Adjugate woh box hai jo main diagonal swap karke aur off-diagonal negate karke milta hai:
adj A = ( d − c − b a ) A − 1 = det A 1 adj A Adjugate and cofactors .
Do ordinary equations,
{ 2 x + y = 5 3 x + 4 y = 6 A ( 2 3 1 4 ) x ( x y ) = b ( 5 6 )
Row rule se check karo: row 1 deta hai 2 x + 1 y = 5 3 x + 4 y = 6
Intuition Yeh packing kyon matter karti haiEk baar A x = b A x = A − 1 b Solving linear systems .
Plain numbers and arithmetic
Matrix as a box of numbers
Matrix multiplication row meets column
Identity matrix do nothing
Determinant as signed area
Inverse undoes the machine
Singular when det is zero
Har question padho, apne dimaag mein jawab do, phir reveal karo.
( a c b d ) Top-left a d
Off-diagonal entries kahan hain? Top-right b c
x Ki
x ek vector (ek arrow) hai, single number nahi.
Matrix ke dono columns geometrically kya represent karte hain? Transformation ke baad basic arrows e ^ 1 e ^ 2
( a c b d ) ( x 2 x 1 ) ( c x 1 + d x 2 a x 1 + b x 2 )
Identity matrix kya hai aur yeh kya karta hai? ( 1 0 0 1 )
det A = a d − b c A
det A = 0 Dono columns parallel hain; area zero ho jaata hai, toh machine undo nahi ho sakti.
Notation A − 1 Woh machine jisme A A − 1 = A − 1 A = I A
( a c b d ) ( d − c − b a )
2 x + y = 5 , 3 x + 4 y = 6 ( 2 3 1 4 ) ( y x ) = ( 6 5 ) , yaani
A x = b .
