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If A = pmatrix y & z & x z & x & y x & y & z pmatrix where x , y , z are integers, is an orthogonal matrix, then what is the value of x^2 + y^2 + z^2 ?

Options

  1. A0
  2. B1
  3. C4
  4. D14

Correct answer

B. 1

Step-by-step solution

For an orthogonal matrix A , the condition A A^T = I must be satisfied. Given A = pmatrix y & z & x z & x & y x & y & z pmatrix , we find A^T = pmatrix y & z & x z & x & y x & y & z pmatrix . Multiplying A and A^T gives: A A^T = pmatrix y & z & x z & x & y x & y & z pmatrix pmatrix y & z & x z & x & y x & y & z pmatrix = pmatrix x^2+y^2+z^2 & xy+yz+zx & xy+yz+zx xy+yz+zx & x^2+y^2+z^2 & xy+yz+zx xy+yz+zx & xy+yz+zx & x^2+y^2+z^2 pmatrix Equating A A^T to the identity matrix I = pmatrix 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1

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