2.6.12 · D1Solving systems using matrix inversion
Linear equations ka poora system secretly EK sentence hai: "ek rule (jise hum A X B ulta chalana hai, aur yeh page — bilkul scratch se — har ek object (A X B
Parent page par jo kuch bhi hai — A X = B X = A − 1 B det A adj ( A ) unme se har ek ko bilkul zero se unpack karta hai, us order mein jisme woh ek doosre par depend karte hain. Agar neeche koi word naya lage, toh aap sahi jagah par hain.
Kisi bhi equation se pehle, hume woh object chahiye jis par sab kuch likha jaata hai.
Ek matrix bas numbers ka ek rectangular grid hoti hai jo rows (aarpaar) aur columns (neeche) mein arranged hoti hai. Hum ise ek capital letter jaise A ( rows ) × ( columns ) A = [ 2 1 3 − 1 ] is a 2 × 2 matrix (2 rows, 2 columns).
Row i j a ij pehle row, phir column . Toh a 21
Ek square matrix woh hoti hai jisme rows ki sankhya barabar ho columns ki sankhya ke (size n × n 2 × 2 3 × 3 2 × 3 square nahi hai.
Abhi hume kyun parwah hai: identity matrix I A − 1 det A adj ( A ) sirf square matrices ke liye defined hai. Ek non-square grid ka koi determinant nahi hota aur koi inverse bhi nahi hota. Isliye parent topic hamesha "n n A
Intuition Grid kyun hoti hai?Ek grid hume bahut saari equations ke coefficients ek saaf jagah mein store karne deti hai, taaki poora system ek single object ban jaaye jise hum ek set of rules se manipulate kar sakein. Figure dekhein: har equation ek row ban jaati hai, har variable apna ek column own karta hai. Jab equations ki count unknowns ki count se match karti hai, woh grid square hoti hai — exactly woh case jahan hum use reverse karne ki umeed kar sakte hain.
YEH TOPIC KO KYUN CHAHIYE: coefficient matrix A X B A X = B
Sirf ek column wali matrix itni special hai ki uska apna naam hai.
Ek column vector ek column wali matrix hoti hai, jaise X = [ x y ] 2 × 1 arrow ki tarah picture karte hain jo origin se start hoti hai aur coordinates ( x , y )
Intuition Ek hi cheez ko dekhne ke do tarikeNumbers ki ek pair [ x y ]
YEH TOPIC KO KYUN CHAHIYE: X B X
Yeh poore topic ka engine hai, isliye hum dheere chalenge.
Definition Dimension-matching ruleAap A m × n X n × 1 tabhi jab inner numbers match karein : A = n X = n outer size m × 1 A ( m × n ) X ( n × 1 ) ⟶ ( m × 1 ) . undefined hota hai — aap ise likh hi nahi sakte. (Size n × n A n × 1 X n × 1
Definition Row-times-column (matrix × vector)Matrix A X A X [ a c b d ] [ x y ] = [ a x + b y c x + d y ] .
HUMNE KYA KIYA: row 1 ( a , b ) ( x , y ) a x + b y c x + d y YEH PATTERN KYUN: kyunki a x + b y YEH KAISA DIKHTA HAI: ek input arrow X A
Definition Matrix × matrix (baad mein identities ke liye chahiye)Wahi rule do poori grids tak extend hota hai: P m × n Q n × p row i j mein entry P i Q j m × p n [ a c b d ] [ p r q s ] = [ a p + b r c p + d r a q + b s c q + d s ] . p = 1 A adj ( A ) A − 1 A
Worked example Machine ko check karoA = [ 2 1 3 − 1 ] X = [ x y ] A X = [ 2 x + 3 y x − y ] . B = [ 5 1 ] exactly 2 x + 3 y = 5 x − y = 1 A X = B hi woh system hai.
Common mistake Column-pehle multiply karnaKyun sahi lagta hai: hum left-to-right padhte hain, isliye A Fix: hamesha left matrix ki row, right matrix ka column . Row-pehle rule hai; column-pehle nonsense deta hai.
Definition Identity matrix (sirf square)Identity matrix I square matrix hai jisme main diagonal (top-left se bottom-right) par 1 0 I = [ 1 0 0 1 ] . I X = X X 1 I
YEH TOPIC KO KYUN CHAHIYE: X = A − 1 B I X I X X I
Definition Inverse matrix (sirf square)A − 1 A square matrix hai jo A A A − 1 A − 1 A = I aur A A − 1 = I . A − 1 A A A − 1 defined hon aur dono same I A n × n koi inverse nahi hota — bas.
Intuition Vending machine ki pictureA X B A − 1 B X A A − 1 I
Chhota − 1 Ordinary numbers mein, 5 − 1 = 5 1 5 × 5 1 = 1 A − 1 A I 1
YEH TOPIC KO KYUN CHAHIYE: X = A − 1 B
Hume ek number chahiye jo bataye ki A − 1 square ke liye quantity hai.
1 × 1 2 × 2 Determinant ek square matrix se compute kiya gaya single number hota hai. Hum ise size ke hisaab se build karte hain:
1 × 1 det [ a ] = a (Hume yeh base case §6 mein chahiye, kyunki 2 × 2 1 × 1 2 × 2 det [ a c b d ] = a d − b c . bade matrices: ek n × n 1 × 1 Determinants .) Non-square grid ka koi determinant nahi hota.
Geometrically 2 × 2 A signed area hai.
Intuition Area = reversibility kyunAgar do column arrows genuinely alag directions mein point karte hain, toh woh real area wala parallelogram banaate hain — machine space ko phelaa deti hai aur undo ki ja sakti hai. Agar arrows SAME line par pad jaate hain, toh parallelogram ka zero area hoga: space ek line par squash ho gaya, bahut saare inputs same output par land karte hain, aur aap kabhi uniquely reverse nahi kar sakte. Woh flat case exactly det A = 0 det A = 0 singular kahlati hai (non-reversible); det A = 0 non-singular kahlati hai.
YEH TOPIC KO KYUN CHAHIYE: A − 1 = det A 1 adj A det A det A = 0 Determinants .
Inverse ko actually compute karne ke liye hume teen linked ideas chahiye. (Poori detail: Cofactors and Minors aur Adjoint and Inverse of a Matrix .)
Definition Minor, cofactor, adjugate
Ek minor M ij row i j ke baad bachti hai. 2 × 2 1 × 1
Ek cofactor C ij = ( − 1 ) i + j M ij
Adjugate adj ( A ) transpose (rows aur columns swap) kiya gaya hai: adj ( A ) ij = C j i
Common mistake Transpose skip karnaKyun sahi lagta hai: cofactor grid aur adjugate mein same numbers hote hain, isliye woh same dikhte hain.
Fix: adj ( A ) diagonal ke paar flip kiya gaya . Hamesha transpose karo; neeche di gayi key identity se verify karo.
Definition Consistent vs inconsistent
Ek system consistent hai agar uska kam se kam ek solution ho.
Woh inconsistent hai agar uska koi solution na ho.
Jab det A = 0 X = A − 1 B
det A = 0 derived , asserted nahi
System A X = B dono sides ko left mein adj ( A ) adj ( A ) A X = adj ( A ) B . adj ( A ) A = ( det A ) I ( det A ) X ( det A ) X = adj ( A ) B . det A = 0 zero column O adj ( A ) B = O
Agar adj ( A ) B = O violate hoti hai — koi bhi X koi solution nahi (inconsistent) .
Agar adj ( A ) B = O X infinitely many solutions, ya koi nahi (aapko substitution se confirm karna hoga).
Qualification: yeh ( adj A ) B 2 × 2 3 × 3 Consistency of Linear Systems aur Gaussian Elimination .
Har item ko ek question ki tarah padho, answer chhupao, aur khud ko test karo.
Matrix kya hoti hai aur hum entry a ij Ek rectangular grid of numbers; a ij i j
Square matrix kya hoti hai, aur yahan yeh kyun matter karti hai? Rows = columns (n × n
Column vector kya hai, aur uske saath kaun si picture jaati hai? Ek one-column matrix; origin se us location tak arrow jise woh list karta hai.
Product A X Jab A X
Matrix ko column vector se multiply kaise karte hain? Matrix ki har row lo, vector ke saath term-by-term multiply karo, aur add karo — left ka row, right ka column.
Do matrices ko multiply kaise karte hain? Entry ( i , j ) i j
A X = B Row i X i b i i
Identity matrix I Ek square matrix, diagonal par 1 0 I X = X
A − 1 Reverse machine jisme A − 1 A = A A − 1 = I
1 × 1 det [ a ] Bas a
det A 2 × 2 det A = 0 Column parallelogram ka signed area; zero ka matlab space flat ho gaya (singular, non-reversible).
Cofactor grid aur adj ( A ) Adjugate, cofactor grid transposed hai (rows aur columns swap kiye gaye).
2 × 2 Chaar cofactors [ d − b − c a ] [ d − c − b a ]
A adj ( A ) = ( det A ) I Diagonal entries det A 0
( det A ) X = adj ( A ) B det A = 0 Agar det A = 0 O adj ( A ) B = O = O
"Consistent" ka kya matlab hai? System mein kam se kam ek solution hai.
