Question
Download Solution PDFमाना A तथा B 2 × 2 का आव्यूह है। तब निम्न में से कौन-से सत्य है?
Answer (Detailed Solution Below)
Detailed Solution
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मान लीजिए
\(A = \left(\begin{array}{cc} a & b \\ c & d \end{array}\right)\)
और
\(B = \left(\begin{array}{cc} x & y \\ z & w \end{array}\right)\)
अब
\(A+B = \left(\begin{array}{cc} a+x & b+y \\ c+z & d+w \end{array}\right)\)
det (A + B ) = (ad - bc) + (xw - yz) + aw + xd - bz - yc . . ..(i)
अब
\(A-B = \left(\begin{array}{cc} a-x & b-y \\ c-z & d-w \end{array}\right)\)
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 \(A=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\) और \(B=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]\)
\(A+B=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\), \(A-B=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\)
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।
Last updated on Jul 19, 2025
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