(ii) Row-Reduction Steps to Find $A^{-1}$

(ii) Row-Reduction Steps to Find $A^{-1}$

Form the augmented matrix $[A \mid I_3]$:

1. $R_3 \to R_3 - R_1$: Eliminate Column 1 in Row 3

   $$\sim \left[\begin{array}{ccc|ccc} 1 & 2 & -1 & 1 & 0 & 0 \\ 0 & -2 & 8 & 0 & 1 & 0 \\ 0 & -2 & 3 & -1 & 0 & 1 \end{array}\right]$$

2. $R_2 \to -\frac{1}{2} R_2$: Leading 1 in Row 2

   $$\sim \left[\begin{array}{ccc|ccc} 1 & 2 & -1 & 1 & 0 & 0 \\ 0 & 1 & -4 & 0 & -\frac{1}{2} & 0 \\ 0 & -2 & 3 & -1 & 0 & 1 \end{array}\right]$$

3. $R_1 \to R_1 - 2R_2, \quad R_3 \to R_3 + 2R_2$: Clear Column 2

   $$\sim \left[\begin{array}{ccc|ccc} 1 & 0 & 7 & 1 & 1 & 0 \\ 0 & 1 & -4 & 0 & -\frac{1}{2} & 0 \\ 0 & 0 & -5 & -1 & -1 & 1 \end{array}\right]$$

4. $R_3 \to -\frac{1}{5} R_3$: Leading 1 in Row 3

   $$\sim \left[\begin{array}{ccc|ccc} 1 & 0 & 7 & 1 & 1 & 0 \\ 0 & 1 & -4 & 0 & -\frac{1}{2} & 0 \\ 0 & 0 & 1 & \frac{1}{5} & \frac{1}{5} & -\frac{1}{5} \end{array}\right]$$

5. $R_1 \to R_1 - 7R_3, \quad R_2 \to R_2 + 4R_3$: Clear Column 3 $\implies [I_3 \mid A^{-1}]$

   $$\sim \left[\begin{array}{ccc|ccc} 1 & 0 & 0 & -\frac{2}{5} & -\frac{2}{5} & \frac{7}{5} \\ 0 & 1 & 0 & \frac{4}{5} & \frac{3}{10} & -\frac{4}{5} \\ 0 & 0 & 1 & \frac{1}{5} & \frac{1}{5} & -\frac{1}{5} \end{array}\right]$$