Question

If \[ A = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix} \] is a non-singular matrix, then find \( A^{-1} \) by elementary row transformations.

Hint:

To find the inverse of a diagonal matrix, simply take the reciprocal of the diagonal elements.

Answer 1

We are given the matrix \( A = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix} \), and we need to find its inverse using elementary row transformations. The inverse of a matrix \( A \) can be found by performing row operations on the augmented matrix \( [A | I] \), where \( I \) is the identity matrix. Step 1: Write the augmented matrix: \[ \left[ \begin{array}{ccc|ccc} x & 0 & 0 & 1 & 0 & 0 \\ 0 & y & 0 & 0 & 1 & 0 \\ 0 & 0 & z & 0 & 0 & 1 \end{array} \right] \] Step 2: Perform row operations to make the diagonal elements equal to 1. We will scale each row by the reciprocal of the corresponding diagonal element. - Row 1: Divide by \( x \), Row 2: Divide by \( y \), Row 3: Divide by \( z \). The result will be: \[ \left[ \begin{array}{ccc|ccc} 1 & 0 & 0 & \frac{1}{x} & 0 & 0 \\ 0 & 1 & 0 & 0 & \frac{1}{y} & 0 \\ 0 & 0 & 1 & 0 & 0 & \frac{1}{z} \end{array} \right] \] Step 3: The right-hand side is the inverse matrix: \[ A^{-1} = \begin{pmatrix} \frac{1}{x} & 0 & 0 \\ 0 & \frac{1}{y} & 0 \\ 0 & 0 & \frac{1}{z} \end{pmatrix} \]