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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Determinants

NDA 2018 · Mathematics · Determinants

What is the determinant of the matrix ( array ccc x & y & y+z z & z...

What is the determinant of the matrix \left(xyy+zzzz+xyzx+y\begin{array}{ccc}x & y & y+z \\ z & z & z+x \\ y & z & x+y\end{array}\right) ?

Options

  1. A(x−y)(y−z)(z−x)(x-y)(y-z)(z-x)
  2. B(x−z)(z−x)(\mathrm{x}-\mathrm{z})(\mathrm{z}-\mathrm{x})
  3. C(y−z)(z−x)(\mathrm{y}-\mathrm{z})(\mathrm{z}-\mathrm{x})
  4. D(z−x)2(x+y+z)(z-x)^2(x+y+z)

Correct answer

D. (z−x)2(x+y+z)(z-x)^2(x+y+z)

Step-by-step solution

\left|xyy+zzxz+xyzx+y\begin{array}{lll}x & y & y+z \\ z & x & z+x \\ y & z & x+y\end{array}\right|=\mathrm{R}_1 \rightarrow \mathrm{R}_1+\mathrm{R}_2+\mathrm{R}_3\left|x+y+zx+y+z2(x+y+z)zxz+xyzx+y\begin{array}{ccc} x+y+z & x+y+z & 2(x+y+z) \\ z & x & z+x \\ y & z & x+y \end{array}\right|=(x+y+z)∣112zxz+xyzx+y∣=(x+y+z)(z+x)2,( replacing z by x)\begin{aligned} & =(x+y+z)\left|\begin{array}{ccc}1 & 1 & 2 \\ z & x & z+x \\ y & z & x+y\end{array}\right| \\ & =(x+y+z)(z+x)^2,(\text{ replacing } z \text{ by } x)\end{aligned}$
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