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If x, y, z are non zero real numbers, then inverse of matrix A= [ array lll x & 0 & 0 0 & y & 0 0 & 0 & z array ] is

Options

  1. Ax y z [ array ccc 1 / x & 0 & 0 0 & 1 / y & 0 0 & 0 & 1 / z array ]
  2. B1 x y z [ array lll 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 array ]
  3. C1 x y z [ array lll x & 0 & 0 0 & y & 0 0 & 0 & z array ]
  4. D[ array ccc 1 / x & 0 & 0 0 & 1 / y & 0 0 & 0 & 1 / z array ]

Correct answer

D. [ array ccc 1 / x & 0 & 0 0 & 1 / y & 0 0 & 0 & 1 / z array ]

Step-by-step solution

The given matrix is a diagonal matrix A = diag (x, y, z) = bmatrix x & 0 & 0 0 & y & 0 0 & 0 & z bmatrix . For a diagonal matrix A = diag (a₁, a₂, a₃) , the inverse A⁻¹ is given by diag (1/a₁, 1/a₂, 1/a₃) , provided all a_i 0 . Since x, y, z are non-zero real numbers, the inverse exists and is calculated as follows: A⁻¹ = bmatrix 1/x & 0 & 0 0 & 1/y & 0 0 & 0 & 1/z bmatrix . Answer: [ array ccc 1 / x & 0 & 0 0 & 1 / y & 0 0 & 0 & 1 / z array ]

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