Determinant of 3×3 matrix — cofactor expansion: Question bank

2.6.8 · D5 · Maths › Matrices & Determinants — Introduction › 3×3 matrix ka Determinant — cofactor expansion

Yeh page ek question bank hai, calculator drill nahi. Neeche har line mein ::: ke baad ek chhota sa answer chhupa hai. Reveal karne se pehle apna jawab zor se bolne ki koshish karo. Goal yeh hai ki woh sneaky misconceptions pakad lo jo numbers crunch karne ke baad bhi bach jaate hain — jaise ki properties aur inverse questions mein test hote hain.

Parent: Determinant of 3×3 — cofactor expansion.

True or false — justify

TRUE or FALSE: Ek hi matrix mein row 1 ke along expand karna aur column 3 ke along expand karna alag-alag determinants de sakta hai.

FALSE — determinant ek matrix ke liye ek fixed number hota hai; har valid row/column expansion sirf ek alag raasta hai usi answer tak pahunchne ka.

TRUE or FALSE: Minor

M_{ij}

aur cofactor

C_{ij}

hamesha equal hote hain.

FALSE — yeh sirf tab equal hote hain jab

i + j

even ho (sign

+ 1

); jab

i + j

odd ho, tab

C_{ij} = - M_{ij}

TRUE or FALSE: Kisi element ka checkerboard sign us element ki value par depend karta hai.

FALSE — sign

(- 1)^{i + j}

sirf position

(i, j)

par depend karta hai, wahan rakhe number par kabhi nahi.

TRUE or FALSE: Agar kisi matrix mein zeros ki ek poori row ho, toh uska determinant zero hota hai.

TRUE — us row ke along expand karo: har term (zero)×(cofactor) hogi, isliye poora sum

0

hoga.

TRUE or FALSE: Hamesha row 1 ke along expand karna chahiye.

FALSE — tum kisi bhi row ya column ke along expand kar sakte ho; jisme zeros zyada hon usse choose karna terms ko khatam kar deta hai aur kaam asaan kar deta hai, isliye row 1 rarely sabse smart choice hoti hai.

TRUE or FALSE: Ek triangular 3×3 matrix (diagonal ke upar ya neeche ke saare entries zero hain) ka determinant uske diagonal entries ke product ke barabar hota hai.

TRUE — zero-heavy row/column ke along baar baar expand karne par sirf diagonal product bachta hai, jaise

det = a_{11} a_{22} a_{33}

TRUE or FALSE: Kisi matrix ki do rows ko swap karne par determinant unchanged rehta hai.

FALSE — ek single row swap determinant ka sign flip kar deta hai (yeh orientation reverse kar deta hai); yeh ek property of determinants hai.

TRUE or FALSE: Agar ek 3×3 matrix ki do rows identical hain, toh determinant zero hota hai.

TRUE — identical rows linearly dependent hoti hain, isliye teen rows flat (coplanar) hoti hain aur zero volume enclose karti hain.

TRUE or FALSE: Cofactor expansion formula sirf tab kaam karta hai jab tum row choose karo, column nahi.

FALSE — column expansion equally valid hai;

det (A) = \sum_{i} a_{ij} C_{ij}

kisi bhi column

j

ke along kaam karta hai.

Spot the error

Ek student likhta hai

det = a_{11} M_{11} + a_{12} M_{12} + a_{13} M_{13}

. Kya galat hai?

Alternating signs missing hain — middle term ko

(- 1)^{1 + 2} = - 1

chahiye, isliye yeh

a_{11} M_{11} - a_{12} M_{12} + a_{13} M_{13}

hona chahiye.

C_{12}

find karne ke liye, ek student row 1 aur column 1 cross out karta hai. Kya mistake hai?

C_{12}

ke liye tum row

1

aur column

2

delete karte ho (element ke indices), column 1 nahi — unhone galat column cross out kiya.

Ek student 2×2 minor compute karta hai

a c b d = a d + b c

. Ise theek karo.

Yeh

a d - b c

hona chahiye (main diagonal minus anti-diagonal); sign subtraction hai, addition nahi — dekho 2×2 determinant.

Ek student column 2 ke along expand karta hai jisme

0, 5, 0

hain, lekin phir bhi teeno cofactors

C_{12}, C_{22}, C_{32}

compute karta hai. Yeh wasteful kyun hai (galat nahi)?

Answer phir bhi sahi hai, lekin

0 \cdot C_{12}

aur

0 \cdot C_{32}

terms vanish ho jaate hain, isliye woh do minors compute karna bilkul wasted effort tha.

Ek student kehta hai "

C_{23}

ka sign

+

hai kyunki 2 aur 3 dono chhote hain." Error pakdo.

Sign

(- 1)^{2 + 3} = (- 1)^{5} = - 1

se aata hai, isliye

C_{23}

negative hai; indices ki size irrelevant hai, sirf yeh matter karta hai ki

i + j

odd hai ya even.

Ek student claim karta hai ki kyunki unke expansion ne negative number diya, toh "zaroor sign error hui hogi." Kya yeh reasoning valid hai?

Nahi — determinants genuinely negative ho sakte hain (negative signed volume ka matlab hai transformation orientation flip karta hai), isliye negative result apne aap mein mistake ka evidence nahi hai.

Ek student

a_{11}

factor out karta hai aur rows 1,2 columns 1,2 ko minor

M_{11}

maanta hai. Kya galat hua?

Minor

M_{11}

woh rows aur columns use karta hai jo row 1 aur column 1 delete karne ke baad bachte hain — yaani rows 2,3 aur columns 2,3, rows 1,2 nahi.

Why questions

Signs

+, -, +

alternate kyun karte hain, sabhi

+

kyun nahi hote?

Yeh orientation encode karte hain: alternating

(- 1)^{i + j}

pattern full determinant sum mein permutations ke sign se aata hai, yeh ensure karta hai ki volumes sahi se add ya subtract hon taaki answer handedness respect kare.

Bahut saare zeros wali row ke along expand karna achha strategy kyun hai?

Har zero element apne cofactor ko multiply karke

0

deta hai, us poore term ko hata deta hai — isliye zyada zeros ka matlab hai kam 2×2 minors actually compute karne padte hain.

Zero determinant ka matlab transformation 3D space ko "collapse" karta hai, kyun?

Zero signed volume ka matlab hai ki teen column (ya row) vectors linearly dependent hain aur ek plane ya line mein lie karte hain, isliye unit cube flat ho jaata hai — yeh ek dimension khota hai aur invert nahi ho sakta (relevant to the inverse).

Hum kisi bhi row ya column ke along expand kar sakte hain aur phir bhi same number milta hai, kyun?

Kyunki yeh saari expansions sirf determinant ki ek underlying permutation-sum definition ki regroup karings hain, jo matrix ke liye ek single fixed value produce karti hai.

Cofactor ke andar sirf ek number ki jagah minor (ek chhota determinant) kyun chahiye?

Kyunki hum ek 3D volume ko ek direction mein slice kar rahe hain; har slice ka contribution khud ek 2D signed area hai, aur ek 2×2 determinant exactly wahi measure karta hai.

3 \times 3

determinant

2 \times 2

determinants se kyun bana hai, na ki entries ke seedhe multiplication se?

3D mein volume is par depend karta hai ki vectors ek doosre ke relative kaise oriented hain, aur sirf alternating minor structure woh capture karta hai; naive entry products orientation aur dependence ignore kar dete hain.

Cross product top row ke along cofactor expansion jaisa kyun dikhta hai?

Cross product ek determinant ke roop mein define hota hai jisme row 1 mein

\hat{i}, \hat{j}, \hat{k}

hote hain, aur us row ke along expand karne par teen components exact alternating signs ke saath milte hain.

Edge cases

Ek 3×3 matrix ka determinant kya hoga jisme ek column entirely zero ho?

Zero — us column ke along expand karo aur har term

0 \times C_{ij}

hogi, isliye poora sum

0

ho jaata hai.

Agar ek 3×3 matrix ke do columns proportional hain (ek doosre ka

k

times hai), toh determinant kya hoga?

Zero — proportional columns linearly dependent hote hain, isliye vectors coplanar hain aur koi volume enclose nahi karte.

3×3 identity matrix ka determinant kya hai, aur geometrically us value ka kya matlab hai?

Yeh

1

hai — identity space ko unchanged chhod deta hai, isliye unit cube apna unit volume aur orientation rakhta hai.

Diagonal matrix

diag (a, b, c)

ke liye cofactor expansion kya deta hai?

Yeh

ab c

deta hai — kisi bhi row ke along expand karo aur sirf diagonal term bachta hai, jo triangular-matrix rule se match karta hai.

Agar ek 3×3 matrix ki ek single row ko factor

k

se scale karo, toh determinant kya hoga?

Yeh

k

se multiply ho jaata hai — woh row expansion ke har surviving term mein ek baar appear hoti hai, isliye

k

factor out karna poore determinant ko

k

se multiply kar deta hai (ek property of determinants).

Agar teen rows satisfy karti hain

row 3 = row 1 + row 2

, toh matrix ka determinant kya hoga?

Zero — rows linearly dependent hain (row 3 doosron se bani hai), isliye woh ek plane mein hain aur koi 3D volume span nahi karti.

Agar ek 3×3 matrix ki har entry same nonzero number ho, toh determinant kya hoga?

Zero — saari rows identical hain, isliye linearly dependent hain, koi volume nahi milta.

Recall One-line survival kit

Element ≠ minor ≠ cofactor; sign sirf position-based hai; zeros tumhare dost hain; aur answer same rehta hai chahe tum kisi bhi row ya column ke along chalo.