Properties — row operations, multiplicativity: Question bank

4.5.22 · D5 · Maths › Linear Algebra (Full) › Properties — row operations, multiplicativity

Quick symbol reminder taaki neeche kuch bhi unexplained na lage:

$det A$ = map $A$ ka signed volume-scaling factor (unit cube ka volume kitna badhta hai, negative agar orientation flip ho). Dekho Volume, Orientation & the Determinant.
Ek matrix $n \times n$ hai (square, $n$ rows aur $n$ columns).
Elementary matrix $E$ = identity matrix jisme ek row operation apply ki gayi ho; isse multiply karna wahi operation hai. Dekho Elementary Matrices.
Singular = not invertible = $det = 0$ . Dekho Invertibility & Singular Matrices.

Spot the error

SE1. "Maine

R_{1} \to \frac{1}{2} R_{1}

se pivot 1 banaya, triangular tak reduce kiya, diagonal multiply ki, aur woh

det A

hai."

Error:

R_{1}

\frac{1}{2}

se scale karne ne

det

\frac{1}{2}

se multiply kiya, isliye final diagonal product

\frac{1}{2} det A

hai;

det A

recover karne ke liye 2 se multiply back karna zaroori hai.

SE2. "

det (2 A) = 2 det A

kyunki scalar ko matrix se bahar nikal sakte hain."

Error: scalar ko ek row se bahar nikala ja sakta hai, lekin

2 A

saari

n

rows ko scale karta hai, har ek se 2 ka ek factor milta hai:

det (2 A) = 2^{n} det A

SE3. "

R_{2} \to R_{2} + 3 R_{1}

ne entries change kiye, isliye determinant bhi change hua."

Error: added piece

3 D (\dots, R_{1}, \dots, R_{1}, \dots)

mein do equal rows hain, isliye woh

0

hai; determinant genuinely unchanged rehta hai chahe matrix alag dikhne lage.

SE4. "

A

singular hai aur

B

invertible hai, isliye

A B

invertible ho sakta hai."

Error:

det (A B) = det A det B = 0 \cdot det B = 0

, isliye

A B

singular hai; ek invertible

B

us collapse ko rescue nahi kar sakta jo

A

ne already cause kiya.

SE5. "Maine ek swap aur ek replacement ki, isliye apna triangular answer

(- 1)^{2} = 1

se multiply karta hoon."

Error: replacements ka factor

1

hota hai, isliye sirf swap matter karta hai — sahi factor

(- 1)^{1} = - 1

hai,

1

nahi.

SE6. "

det (A^{3}) = 3 det A

Error: matrix ki powers map ko teen baar compose karti hain, isliye scalings multiply hoti hain:

det (A^{3}) = (det A)^{3}

, na ki

3 det A

SE7. "

det E = 0

replacement matrix ke liye kyunki woh identity nahi hai."

Error: ek replacement elementary matrix triangular hoti hai jisme diagonal par saare 1 hote hain, isliye

det E = 1

—

I

se alag hona determinant ko zero nahi banata.

Why questions

WHY1. Do rows swap karne par sign kyun flip hota hai, bina koi naya axiom add kiye?

Dono slots mein equal rows

u + v

dene par

0

milta hai; linearity se expand karne par

D (u, v) + D (v, u) = 0

bachta hai, isliye

D (v, u) = - D (u, v)

— sign flip "equal rows give 0" plus multilinearity se forced hai. Dekho Multilinear Alternating Forms.

WHY2. Gaussian elimination ke liye replacement "safe" operation kyun hai?

Kyunki yeh

det

ko exactly unchanged chodti hai, tum entries clear karke echelon form tak pahunch sakte ho bina koi correction factor track kiye.

WHY3. Ek triangular matrix ka determinant sirf diagonal ka product kyun hota hai?

Pehle column ke along cofactor expansion har stage par sirf ek surviving nonzero term chodti hai, jo recursively diagonal entries ko saath multiply karte rehte hain. Dekho Determinant — Definition (cofactor / Leibniz).

WHY4. Multiplicativity

det (A B) = det A det B

geometrically kyun hold karta hai?

B

volume ko

det B

se scale karta hai, phir

A

us result ko

det A

se scale karta hai; do maps compose karna scaling factors ko multiply karta hai, aur sign flips (orientation) bhi usi tarah multiply hoti hain.

WHY5. Multiplicativity ka algebraic proof invertible aur singular cases mein kyun split hota hai?

Sirf invertible

A

hi elementaries

E_{1} \dots E_{m}

ke product ke roop mein factor hota hai; singular case ke liye alag argument chahiye, jisme use kiya jaata hai ki

det A = 0

aur

A B

rank deficiency inherit karta hai isliye

det (A B) = 0

bhi.

WHY6.

det (A^{T}) = det A

kyun har row rule ko columns par apply karne deta hai?

A

par ek column operation,

A^{T}

par ek row operation hai; kyunki transpose karne se determinant nahi badalta,

det

par effect identical hota hai.

WHY7.

det (A^{- 1}) = 1/ det A

kyun zaroori hai aur yeh invertible matrices ka nonzero determinant kyun prove karta hai?

det A \cdot det (A^{- 1}) = det I = 1

se, product

1

hai, isliye koi bhi factor

0

nahi ho sakta — hence

det A \neq = 0

inverse exist karne ke liye necessary hai.

Edge cases

EC1.

n \times n

zero matrix ka

det

kya hai, aur kyun?

0

== — har row zero row hai, aur zero row

det = 0

force karta hai (use

k

se scale karne par woh change nahi hoti lekin

det

k

se multiply karna padta hai).

EC2. Agar

A

ki do rows identical hain, toh

det A

kya hai?

0

== directly alternating property se — determinant ka defining rule yahi hai ki equal rows

0

dete hain.

EC3. Agar ek row doosri ki scalar multiple hai, toh kya

det A

zero hai?

Haan. Ek replacement

R_{i} \to R_{i} - c R_{j}

(

det

unchanged) row

i

ko all zeros bana sakta hai, aur zero row

det = 0

deta hai.

EC4. Ek

1 \times 1

matrix

[a]

ke liye, kya

det (k A) = k^{n} det A

phir bhi hold karta hai?

Haan, trivially —

n = 1

ke saath yeh padhta hai

k a = k^{1} \cdot a

, woh ek case jahan

det (k A) = k det A

sahi hai.

EC5. Ek permutation matrix ka

det

kya hai jo rows ko odd number of times swap karta hai?

- 1

== — odd swaps mein se har ek identity ke determinant

1

- 1

se multiply karta hai, result

(- 1)^{odd} = - 1

EC6. Agar

A

ek real matrix hai jisme

det A < 0

, toh sign geometrically kya mean karta hai?

Map

A

orientation reverse karta hai (shape ko mirror ki tarah flip karta hai) jabki volume ko

∣ det A ∣

se scale karta hai. Dekho Volume, Orientation & the Determinant.

EC7. Agar

det A = 1

aur

λ

eigenvalues par range karta hai, toh unpar kya constraint aati hai?

Unka product $1$ hai, kyunki

det A = \prod λ_{i}

; isliye volume-preserving maps ke eigenvalues ka product

1

hota hai. Dekho Eigenvalues — det as product of eigenvalues.

EC8. Kya ek replacement operation

det

change kar sakti hai agar accidentally

i = j

use ho (apni hi row ka multiple add karna)?

Rule mein

i \neq = j

zaroori hai;

k R_{i}

R_{i}

mein khud add karna actually ek scaling

R_{i} \to (1 + k) R_{i}

hai, jo

det

(1 + k)

se multiply karta hai — isliye index dhyan se dekho.

Question bank — Properties — row operations, multiplicativity

True or false — justify

Spot the error

Why questions

Edge cases