The matrix $B$ given in the image is:

$B = \begin{pmatrix} 2 & 1 & 3 \\ -1 & 1 & 2 \\ 1 & -1 & -2 \end{pmatrix}$

To find the inverse of matrix $B$ using elementary row operations, we set up an augmented matrix with $B$ on the left and the identity matrix on the right:

$\left( B | I \right) = \begin{pmatrix} 2 & 1 & 3 & | & 1 & 0 & 0 \\ -1 & 1 & 2 & | & 0 & 1 & 0 \\ 1 & -1 & -2 & | & 0 & 0 & 1 \end{pmatrix}$

Using the Gauss-Jordan elimination method, we will perform row operations to transform $B$ on the left into the identity matrix. The resulting matrix on the right will be $B^{-1}$, the inverse of $B$, if $B$ is invertible.

Would you like me to proceed with each step of the row operations for clarity?

Related Questions

1. What are elementary row operations, and how are they used to find a matrix inverse?
2. How do we verify that a matrix is invertible?
3. Can you explain the Gauss-Jordan elimination process in detail?
4. What if matrix $B$ is not invertible; what would that mean for this problem?
5. How does matrix multiplication relate to finding the inverse?

Tip

Always verify that a matrix is invertible by checking if its determinant is non-zero before attempting to find the inverse.