Question 57 of 58intermediate🔧 ApplyMCQ1 mark

If x, y, z are nonzero real numbers, then the inverse of matrix A = [x000y000z]\begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix} is

Exercise: Miscellaneous Exercises on Chapter 4 | Q: 8 | (Page 25)
For More Understanding

Explanation

The given matrix A is a diagonal matrix. For a diagonal matrix, the inverse is obtained by taking the reciprocal of each diagonal element. Since A = diag(x, y, z), its inverse is A−1A^{-1} = diag(1/x, 1/y, 1/z) = diag(x−1x^{-1}, y−1y^{-1}, z−1z^{-1}). This can also be verified using the formula A−1A^{-1} = adj(A)/|A|. The determinant |A| = xyz, and the adjoint of this diagonal matrix has yz, xz, xy on the diagonal, giving A−1A^{-1} = (1/xyz) × diag(yz, xz, xy) = diag(1/x, 1/y, 1/z).

Solution Steps

  1. Step 1: Identify that A is a diagonal matrix with elements x, y, z on the diagonal.

  2. Step 2: For a diagonal matrix, the inverse is obtained by taking reciprocal of each diagonal element.

  3. Step 3: Therefore, A−1A^{-1} = diag(1/x, 1/y, 1/z) = diag(x−1x^{-1}, y−1y^{-1}, z−1z^{-1}).

  4. Step 4: This matches option (A).

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