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NDA 1 2025 Maths, question 30

Topic: Matrices and determinants

Q30

If A=[xyzyzxzxy]A = \begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y \end{bmatrix}, where x,y,zx,y,z are integers, is an orthogonal matrix, then what is A2A^{2} equal to?

  1. (a) Null matrix
  2. (b) Identity matrixCorrect answer
  3. (c) AA
  4. (d) −A- A

Solution.

Answer: (b) Identity matrix

  1. The displayed matrix is symmetric, so AT=AA^{T} = A.
  2. Orthogonality gives AAT=IAA^{T} = I.
  3. Replacing ATA^{T} by AA yields A2=IA^{2} = I, the identity matrix.