To find the inverse of the matrix $B = \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix}$ using elementary transformations, we'll apply row operations to convert the matrix into the identity matrix, while performing the same operations on the identity matrix. The result will be the inverse matrix.

Step-by-step process:

1. Write down the augmented matrix: 4 & 7 & 1 & 0 \\ 2 & 6 & 0 & 1 \\ \end{array}\right]$$ Here, the first part is the matrix $$B$$, and the second part is the identity matrix $$I$$.
2. Make the pivot element in the first row and first column equal to 1. To do this, divide the first row by 4: $R_1 \rightarrow \frac{1}{4} R_1$ 1 & \frac{7}{4} & \frac{1}{4} & 0 \\ 2 & 6 & 0 & 1 \\ \end{array}\right]$$
3. Make the element below the pivot (in the second row, first column) equal to 0. To do this, subtract $2 \times$ the first row from the second row: $R_2 \rightarrow R_2 - 2R_1$ 1 & \frac{7}{4} & \frac{1}{4} & 0 \\ 0 & \frac{5}{2} & -\frac{1}{2} & 1 \\ \end{array}\right]$$
4. Make the pivot in the second row and second column equal to 1. To do this, multiply the second row by $\frac{2}{5}$: $R_2 \rightarrow \frac{2}{5} R_2$ 1 & \frac{7}{4} & \frac{1}{4} & 0 \\ 0 & 1 & -\frac{1}{5} & \frac{2}{5} \\ \end{array}\right]$$
5. Make the element above the pivot (in the first row, second column) equal to 0. To do this, subtract $\frac{7}{4} \times$ the second row from the first row: $R_1 \rightarrow R_1 - \frac{7}{4} R_2$ 1 & 0 & \frac{9}{20} & -\frac{7}{20} \\ 0 & 1 & -\frac{1}{5} & \frac{2}{5} \\ \end{array}\right]$$

Now the matrix on the left is the identity matrix, and the matrix on the right is the inverse of $B$.

Thus, the inverse of $B$ is:

\frac{9}{20} & -\frac{7}{20} \\ -\frac{1}{5} & \frac{2}{5} \end{bmatrix}$$ ### Do you have any questions about this process, or would you like further details? Here are some related questions for you: 1. How can you check if a matrix is invertible using its determinant? 2. What happens if you multiply a matrix by its inverse? 3. How can you use row reduction to solve a system of linear equations? 4. What is the general process of finding the inverse of a matrix using Gaussian elimination? 5. How can you compute the inverse of a 3x3 matrix using elementary row operations? **Tip**: Always check the determinant of a matrix before trying to find its inverse. If the determinant is zero, the matrix does not have an inverse.