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The image contains a proof that a square matrix \( A \) is invertible if and only if \( \det(A) \neq 0 \), involving the relationship between the determinant of the matrix and its reduced row echelon form \( R \). The elementary steps in the proof go as follows:

1. The goal is to show that \( \det(A) \neq 0 \) if and only if \( \det(R) \neq 0 \).
2. \( E_1, E_2, \dots, E_r \) are the elementary matrices corresponding to the row operations that reduce \( A \) to \( R \).
3. Therefore, \( E_r E_{r-1} \dots E_1 A = R \), meaning these matrices transform \( A \) into its reduced form \( R \).
4. Taking the determinant of both sides:
   \[
   \det(E_r) \det(E_{r-1}) \dots \det(E_1) \det(A) = \det(R)
   \]
5. Since the determinants of the elementary matrices \( E_i \) are non-zero (depending on the type of row operation), we can conclude that \( \det(A) \neq 0 \) if and only if \( \det(R) \neq 0 \).
6. Assuming \( A \) is invertible, this implies that \( \det(A) \neq 0 \), and hence \( R = I_n \), where \( I_n \) is the identity matrix.
7. If \( \det(A) = 0 \), then \( \det(R) = 0 \), meaning \( R \) cannot be invertible (it would have a row of zeros).
8. Therefore, \( A \) is invertible if and only if \( R = I_n \) and \( \det(A) \neq 0 \).

**The fact that \( \det(EA) = \det(E)\det(A) \) for an elementary matrix \( E \) is used in step 5, where the determinants of both sides of the equation \( E_r E_{r-1} \dots E_1 A = R \) are taken.**

Would you like a further breakdown of any specific part of this proof? Here are five follow-up questions that could deepen understanding:

1. How do different types of elementary row operations affect the determinant of a matrix?
2. Why does the determinant of an invertible matrix need to be non-zero?
3. Can you explain how the reduced row echelon form \( R \) is connected to matrix invertibility?
4. How do you compute the determinant of an elementary matrix?
5. What happens to the determinant of a matrix when a row of zeros appears?

**Tip:** When computing determinants, remember that row swaps change the sign of the determinant, while multiplying a row by a scalar multiplies the determinant by that scalar.

Prove that a square matrix A is invertible if and only if det(A) ≠ 0 using elementary matrices and the reduced row echelon form.

Learn how to prove that a square matrix A is invertible if and only if det(A) ≠ 0 using elementary matrices and reduced row echelon form (RREF). This proof covers key concepts in linear algebra, including determinants, matrix inversion, and the role of elementary row operations.

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