2.6.10
Ek matrix A machine hai jo vectors ko transform karti hai: x ↦ A x Inverse A − 1 undoes karti hai — output ko wapas input pe bhejti hai. Agar A A − 1 nahi undo kar sakte: jab A squash kar de (2D ko ek line mein flatten kar de). Yeh squashing determinant se measure hoti hai. Koi area nahi ⟹ information lost ⟹ koi inverse nahi.
Definition Inverse matrixEk square matrix A A − 1 A A − 1 = A − 1 A = I I hota hai use invertible (ya non-singular ) kehte hain. Agar koi inverse exist nahi karta, toh A singular hai.
Ek 2 × 2 A = ( a c b d )
Hum chahte hain A − 1 = ( p r q s ) A A − 1 = I
Yahan se kyun shuru karein? Kyunki "inverse" defined hi A A − 1 = I
( a c b d ) ( p r q s ) = ( 1 0 0 1 )
Multiply karne par chaar equations milti hain:
a p + b r c p + d r = 1 ( 1 ) = 0 ( 3 ) a q + b s c q + d s = 0 ( 2 ) = 1 ( 4 ) p , r Yeh do kyun? Inme sirf p , r
(3) se: c p + d r = 0 ⇒ r = − d c p d = 0 a p + b ( − d c p ) = 1 ⇒ p d a d − b c = 1 ⇒ p = a d − b c d
Phir r = − a d − b c c
q , s q = a d − b c − b , s = a d − b c a
Quantity a d − b c — ise determinant det A
Intuition "det = 0 pe inverse kyun nahi" — intuitiondet A = a d − b c signed area hai jo A det A = 0 A
Worked example Example 1 — ek clean invertible matrixA = ( 2 3 1 4 )
Step 1: determinant. det A = ( 2 ) ( 4 ) − ( 1 ) ( 3 ) = 8 − 3 = 5 Pehle kyun? Agar yeh 0 hota toh hum turant ruk jaate — koi inverse nahi.
Step 2: adjugate. Diagonal swap karo, off-diagonal negate karo: ( 4 − 3 − 1 2 ) Kyun? Yahi derivation ka pattern hai.
Step 3: det se divide karo.
A − 1 = 5 1 ( 4 − 3 − 1 2 )
Step 4: verify karo A A − 1 = I 5 1 ( 2 3 1 4 ) ( 4 − 3 − 1 2 ) = 5 1 ( 5 0 0 5 ) = I . Verify kyun karein? Forecast-then-verify — sign slips ke against sasta insurance.
Worked example Example 2 — inverse se linear system solve karnaSolve karo { 2 x + y = 5 3 x + 4 y = 6 A x = b A b = ( 6 5 )
Step 1: x = A − 1 b Kyun? A x = b A − 1 A − 1 A x = x
Step 2: x = 5 1 ( 4 − 3 − 1 2 ) ( 6 5 ) = 5 1 ( − 15 + 12 20 − 6 ) = 5 1 ( − 3 14 ) = ( − 0.6 2.8 ) .
Check: 2 ( 2.8 ) + ( − 0.6 ) = 5.6 − 0.6 = 5 3 ( 2.8 ) + 4 ( − 0.6 ) = 8.4 − 2.4 = 6
Worked example Example 3 — ek singular matrixA = ( 2 1 4 2 ) det A = ( 2 ) ( 2 ) − ( 4 ) ( 1 ) = 4 − 4 = 0. Conclusion: No inverse exists . Dekho row 2, row 1 ka aadha hai — rows/columns parallel hain, area collapse ho gaya. Formula mein 0
Common mistake "Inverse bas har entry ka reciprocal hai."Kyun sahi lagta hai: Ek single number ke liye, x − 1 = 1/ x Kyun galat hai: Matrix multiplication rows aur columns ko mix karta hai; ise undo karne ke liye swap/negate/divide structure chahiye, entrywise flips nahi. ( 2 3 1 4 ) ( 1/2 1/3 1 1/4 ) I multiply nahi karta.
Fix: Hamesha d e t A 1 ( d − c − b a )
Common mistake "Invert karne se pehle determinant check karna bhool gaye."Kyun sahi lagta hai: Aap formula apply karne ke liye eager hote ho.
Fix: det A pehle compute karo. Agar 0
Common mistake "Diagonal negate kar di aur off-diagonal swap kar di (ulta!)."Kyun sahi lagta hai: "Kuch negate hota hai, kuch swap hota hai" — kaunsa kya hai yeh mix ho jaata hai.
Fix: ==Swap karo main diagonal, negate karo off-diagonal.== Neeche wala mnemonic ise pakka kar deta hai.
Mnemonic "Corners swap karo, middle flip karo, det share karo."
Corners swap karo: a ↔ d Middle flip karo: b c Det share karo: sab kuch a d − b c
Recall Feynman: 12-saal ke bachche ko explain karoSocho ek machine hai jo ek shape ko stretch aur tilt karti hai. Inverse machine shape ko exactly waisa wapas rakh deti hai jaisa pehle tha. 2×2 grid of numbers ke liye "undo" machine banane ke liye: main diagonal ke dono numbers swap karo, doosre dono pe minus signs lagao, aur sab kuch determinant naam ke ek special number se shrink karo. Agar woh special number zero ho, toh machine ne shape ko ek line mein flatten kar diya aur itni information kho di ki koi "undo" machine kabhi use rebuild nahi kar sakti — jaise paint ko un-mix karna. Toh: zero determinant = koi undo nahi.
A − 1 A A − 1 = A − 1 A = I
( a c b d ) a d − b c 1 ( d − c − b a )
2×2 matrix ( a c b d ) a d − b c
2×2 matrix ka inverse kab NAHI hota? Jab det A = a d − b c = 0
det A = 0 Columns parallel hain; map area ko zero kar deta hai, information kho jaati hai.
Inverse recipe mein main diagonal entries a , d Unhe swap kiya jaata hai.
Inverse recipe mein b c Unhe negate kiya jaata hai (sign flip hoti hai).
Inverse use karke A x = b Matrix inverse ke liye "har entry ka reciprocal" kyun galat hai? Kyunki matrix multiplication rows aur columns mix karta hai; yeh I
( a c b d ) ( d − c − b a ) A − 1 = d e t A 1 adj A
swap diagonal negate off-diagonal
A times A inverse equals I
