We will use the method of row reduction (Gaussian elimination) to find the inverse of the given $2 \times 2$ matrix:

$A = \begin{bmatrix} 5 & 1 \\ 9 & 2 \end{bmatrix}.$

Step 1: Augment the matrix $A$ with the identity matrix $I$:

$\begin{bmatrix} 5 & 1 & 1 & 0 \\ 9 & 2 & 0 & 1 \end{bmatrix}.$

Step 2: Perform row operations to transform $A$ into $I$.

Row 1: Divide the first row by 5 (make $A[1, 1] = 1$):

$\begin{bmatrix} 1 & \frac{1}{5} & \frac{1}{5} & 0 \\ 9 & 2 & 0 & 1 \end{bmatrix}.$

Row 2: Subtract $9 \times \text{Row 1}$ from Row 2:

$\begin{bmatrix} 1 & \frac{1}{5} & \frac{1}{5} & 0 \\ 0 & \frac{1}{5} & -\frac{9}{5} & 1 \end{bmatrix}.$

Row 2: Multiply Row 2 by 5 (make $A[2, 2] = 1$):

$\begin{bmatrix} 1 & \frac{1}{5} & \frac{1}{5} & 0 \\ 0 & 1 & -9 & 5 \end{bmatrix}.$

Row 1: Subtract $\frac{1}{5} \times \text{Row 2}$ from Row 1:

$\begin{bmatrix} 1 & 0 & \frac{2}{5} & -\frac{1}{5} \\ 0 & 1 & -9 & 5 \end{bmatrix}.$

The augmented matrix now looks like: $\begin{bmatrix} 1 & 0 & \frac{2}{5} & -\frac{1}{5} \\ 0 & 1 & -9 & 5 \end{bmatrix}.$

Step 3: Extract the inverse matrix:

The inverse of $A$ is: $A^{-1} = \begin{bmatrix} \frac{2}{5} & -\frac{1}{5} \\ -9 & 5 \end{bmatrix}.$

Step 4: Verify by multiplication:

Multiply $A$ and $A^{-1}$:

\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.$$ This confirms that the inverse is correct. --- Do you have any questions or need further clarification? Here are five related problems to explore: 1. What if one row of the matrix $$A$$ was zero? How does it affect the inverse? 2. Can you use a determinant to check if a matrix is invertible before row reduction? 3. Find the inverse of a $$3 \times 3$$ matrix using row reduction. 4. Solve a system of equations using the inverse of a matrix. 5. What happens if two rows in $$A$$ are proportional? **Tip:** Always verify your inverse matrix by multiplying it with the original matrix to ensure the result is the identity matrix.