A = \(\begin{bmatrix} 1& -1 &0 \\ 3& 2 & -1 \end{bmatrix}\)  और B = \(\begin{bmatrix} 1 \\ 3\\ 5 \end{bmatrix}\) है, तो  (AB)T का मान ज्ञात कीजिए। 

  1. \(\begin{bmatrix} -2 \\ 4 \end{bmatrix}\)
  2. \(\begin{bmatrix} -2 & 4 \end{bmatrix}\)
  3. \(\begin{bmatrix} 2 \\ -4 \end{bmatrix}\)
  4. \(\begin{bmatrix} -2 &- 4 \end{bmatrix}\)

Answer (Detailed Solution Below)

Option 2 : \(\begin{bmatrix} -2 & 4 \end{bmatrix}\)
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NDA 01/2025: English Subject Test
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संकल्पना:

आव्यूह का पक्षांतर:

साधारण आव्यूह के पंक्तियों और स्तंभों को एक-दूसरे से परिवर्तित करके प्राप्त नए आव्यूह को आव्यूह का पक्षांतर कहा जाता है। 

इसे A′ या Aद्वारा दर्शाया गया है। 

गणना:

दिया गया है A = \(\begin{bmatrix} 1& -1 &0 \\ 3& 2 & -1 \end{bmatrix}\) और B = \(\begin{bmatrix} 1 \\ 3\\ 5 \end{bmatrix}\)

AB = \(\begin{bmatrix} 1& -1 &0 \\ 3& 2 & -1 \end{bmatrix}\) × \(\begin{bmatrix} 1 \\ 3\\ 5 \end{bmatrix}\)

AB = \(\begin{bmatrix} 1-3+0 \\ 3+6-5 \end{bmatrix}\)

AB = \(\begin{bmatrix} -2 \\ 4 \end{bmatrix}\)

∴ (AB)T = \(\begin{bmatrix} -2 & 4 \end{bmatrix}\)

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