Question
Download Solution PDFरैखिक समीकरण निकाय kx + y + z = 1, x + ky + z = 1 और x + y + kz = 1 का एकमात्र हल होगा, यदि
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना
माना कि समीकरणों की प्रणाली निम्न है,
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3
⇒ AX = B
⇒ X = A-1 B =
⇒ यदि det (A) ≠ 0 है, तो प्रणाली विशिष्ट हल वाली संगत प्रणाली है।
गणना:
रैखिक समीकरण की दी गयी प्रणाली kx + y + z = 1, x + ky + z = 1 और x + y + kz = 1 हैं।
माना कि A =
det (A) = |A| = k (k2 – 1) – 1(k -1) + 1 (1 – k)
⇒ |A| = k3 – k – k + 1 + 1 – k = k3 – 3k + 2
विशिष्ट हल के लिए,
det (A) ≠ 0
⇒ k3 – 3k + 2 ≠ 0
⇒ (k – 1) (k2 + k - 2) ≠ 0
⇒ (k – 1) (k – 1) (k + 2) ≠ 0
∴ k ≠ 1 और k ≠ -2
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