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NDA 2015
Exam
NDA
Year
2015
Subject
Mathematics
Chapter
Determinants

NDA 2015 · Mathematics · Determinants

If a, b, c are real numbers, then the value of the determinant |...

If a,b,ca, b, c are real numbers, then the value of the determinant \left|1−aa−b−cb+c1−bb−c−ac+a1−cc−a−ba+b\begin{array}{lll} 1-a & a-b-c & b+c \\ 1-b & b-c-a & c+a \\ 1-c & c-a-b & a+b \end{array}$$\right|

Options

  1. A0
  2. B(a−b)(b−c)(c−a)(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c})(\mathrm{c}-\mathrm{a})
  3. C(a+b+c)2(a+b+c)^2
  4. D(a+b+c)3(a+b+c)^3

Correct answer

A. 0

Step-by-step solution

\left|1−aa−b−cb+c1−bb−c−ac+a1−cc−a−ba+b\begin{array}{lll} 1-a & a-b-c & b+c \\ 1-b & b-c-a & c+a \\ 1-c & c-a-b & a+b \end{array}\right|$ apply \mathrm{C}_2 \rightarrow \mathrm{C}_2+\mathrm{C}_3\left|\begin{array}{lll} 1-a & a & b+c \\ 1-b & b & c+a \\ 1-c & c & a+b \end{array}\right|$ apply $\mathrm{C}_1$ \rightarrow $\mathrm{C}_1+\mathrm{C}_2\left|\begin{array}{lll} 1 & \mathrm{a} & \mathrm{~b}+\mathrm{c} \\ 1 & \mathrm{~b} & \mathrm{c}+\mathrm{a} \\ 1 & \mathrm{c} & \mathrm{a}+\mathrm{b} \end{array}$\right|$ apply \mathrm{C}_3 \rightarrow \mathrm{C}_2+\mathrm{C}_3(\mathrm{a}$+$\mathrm{b}+\mathrm{c})\left|\begin{array}{lll} 1 &\mathrm{a}& 1 \\ 1 &\mathrm{~b}& 1 \\ 1 &\mathrm{c}& 1 \end{array}$\right|=0
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