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माना A

तथा B 2 × 2 का आव्यूह है। तब निम्न में से कौन-से सत्य है?

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  1. det(A + B) + det(A - B) = det A + det B
  2. det(A + B) + det(A - B) = 2det A - 2det B
  3. det(A + B) + det(A - B) = 2det A + 2det B
  4. det(A + B) - det(A - B) = 2det A - 2det B

Answer (Detailed Solution Below)

Option 3 : det(A + B) + det(A - B) = 2det A + 2det B
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गणना:

मान लीजिए

और

अब

det (A + B ) = (ad - bc) + (xw - yz) + aw + xd - bz - yc . . ..(i)

अब

det (A - B) = ( ad - bc) + (xw - yz) - xd - aw + bz + yc . . . . (ii)

अब समीकरणों (i) और (ii) से, हमें मिलता है

det (A + B ) + det(A - B) = 2 det(A) + 2 det (B)

इसलिए विकल्प (3) सही है।

Alternate Method 
मान लीजिए और

,

तब (1) ⇒ 1 + 1 = 1 + 0 (→ ←) - विकल्प (1) गलत है

(4) ⇒ 1 - 1 = 2 - 0 (→ ←) विकल्प (4) गलत है

मान लीजिए A = I2x2, B = -I2x2 तब A + B = O2x2, A - B = 2l2x2

(2) ⇒ 0 + 2 = 2 - 2 (→ ←) विकल्प (2) गलत है।

इसलिए, विकल्प (3) सही है।

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