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NDA 1 2025 Maths, question 24

Topic: Matrices and determinants

Q24

If A=[yzxzxyxyz]A = \begin{bmatrix} y & z & x \\ z & x & y \\ x & y & z \end{bmatrix}, where x,y,zx,y,z are integers, is an orthogonal matrix, then what is the value of x2+y2+z2x^{2} + y^{2} + z^{2}?

  1. (a) 0
  2. (b) 1Correct answer
  3. (c) 4
  4. (d) 14

Solution.

Answer: (b) 1

  1. An orthogonal matrix satisfies AAT=IAA^{T} = I.
  2. In particular, the squared length of each row is 1.
  3. The first row therefore gives y2+z2+x2=1y^{2} + z^{2} + x^{2} = 1.