To find the inverse of a matrix using elementary row operations, we augment the matrix with the identity matrix and perform row operations until the original matrix is transformed into the identity matrix. The operations performed on the identity matrix will give us the inverse.

Given matrix $A$:

-1 & -3 & 3 & -1 \\ 1 & 1 & -1 & 0 \\ 2 & -5 & 2 & -3 \\ -1 & 1 & 0 & 1 \\ \end{bmatrix}$$ ### Step-by-Step Procedure 1. **Form the augmented matrix** by combining $$A$$ with the identity matrix $$I$$. \[ \left[ A | I \right] = \left[ \begin{array}{cccc|cccc} -1 & -3 & 3 & -1 & 1 & 0 & 0 & 0 \\ 1 & 1 & -1 & 0 & 0 & 1 & 0 & 0 \\ 2 & -5 & 2 & -3 & 0 & 0 & 1 & 0 \\ -1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ \end{array} \right] \] 2. **Apply row operations** to convert the left side of the augmented matrix into the identity matrix. I'll perform the operations and compute the results step by step. Let's compute this now.The inverse of the given matrix $$A$$ is: $$A^{-1} = \begin{bmatrix} 0 & 2 & 1 & 3 \\ 1 & 1 & -1 & -2 \\ 1 & 2 & 0 & 1 \\ -1 & 1 & 2 & 6 \\ \end{bmatrix}$$ This was found using elementary row operations. Would you like further details on the steps involved in the row operations? Here are 5 related questions: 1. What are the conditions under which a matrix has an inverse? 2. How do elementary row operations help in finding the inverse of a matrix? 3. Can every square matrix be inverted? Why or why not? 4. How would the process change for a non-square matrix? 5. What is the computational complexity of inverting a matrix using this method? **Tip:** When performing matrix operations by hand, carefully track each row operation to avoid mistakes.