The image provided contains a proof of the theorem: "A square matrix $A$ is invertible if and only if $\det(A) \neq 0$." The proof proceeds in steps, involving elementary matrices and the properties of determinants.

In particular, the question at the bottom asks:

"In which line of the proof is the fact that $\det(EA) = \det(E) \det(A)$ for an elementary matrix $E$ used?"

Looking at the proof:

• This specific fact is used in line 5, where the determinants of both sides are taken, and it shows:

  $\det(E_r \cdot \ldots \cdot E_1 A) = \det(R)$

  By expanding the determinant of the left-hand side, this becomes:

  $\det(E_r) \det(E_{r-1}) \cdots \det(E_1) \det(A) = \det(R)$

  This relies on the fact that for elementary matrices $E$, the determinant of their product with $A$ satisfies $\det(EA) = \det(E)\det(A)$.

Thus, the answer to the question is line 5.

Would you like further clarification or details on any part of the proof?

Here are 5 related questions that expand on this information:

1. What are elementary matrices and how do they affect the determinant of a matrix?
2. Why is the determinant being non-zero necessary for matrix invertibility?
3. How do row operations relate to the process of finding the reduced row echelon form?
4. What properties of determinants are used in proving invertibility?
5. How do elementary matrices help in reducing a matrix to its echelon form?

Tip: When working with matrix proofs, always keep track of how elementary row operations affect both the matrix and its determinant!