Prove that an n × n matrix A is invertible if and only if it can be expressed as a
product of elementary matrices.

### Theorem:
An \( n \times n \) matrix \( A \) is invertible **if and only if** it can be expressed as a product of elementary matrices.

#### Proof:

### (1) **If \( A \) is invertible, then it can be expressed as a product of elementary matrices.**

We start by assuming that \( A \) is invertible. This means that there exists a matrix \( B \) such that:
\[
A B = I_n
\]
where \( I_n \) is the identity matrix of size \( n \times n \).

To prove that \( A \) can be written as a product of elementary matrices, we need to recall that **elementary row operations** can transform a matrix into its row-reduced echelon form (RREF). Importantly, every elementary row operation corresponds to multiplying the matrix by an elementary matrix.

Since \( A \) is invertible, we know that its row-reduced echelon form must be the identity matrix \( I_n \). Therefore, we can perform a finite sequence of elementary row operations to transform \( A \) into \( I_n \). Let these operations be represented by elementary matrices \( E_1, E_2, \dots, E_k \), such that:

\[
E_k E_{k-1} \dots E_2 E_1 A = I_n
\]

Multiplying both sides of this equation by \( A^{-1} \), we get:
\[
A = E_1^{-1} E_2^{-1} \dots E_k^{-1}
\]

Since the inverse of an elementary matrix is also an elementary matrix, this shows that \( A \) can be written as a product of elementary matrices.

### (2) **If \( A \) can be expressed as a product of elementary matrices, then \( A \) is invertible.**

Now, assume that \( A \) can be expressed as a product of elementary matrices:
\[
A = E_1 E_2 \dots E_k
\]
where each \( E_i \) is an elementary matrix.

Each elementary matrix \( E_i \) is invertible, with its inverse also being an elementary matrix. Therefore, the product of invertible matrices is invertible. This means that \( A \), as the product of invertible matrices, is itself invertible.

Thus, if \( A \) can be expressed as a product of elementary matrices, it must be invertible.

### Conclusion:
We have shown both directions of the proof:
1. If \( A \) is invertible, then it can be expressed as a product of elementary matrices.
2. If \( A \) can be expressed as a product of elementary matrices, then it is invertible.

Hence, \( A \) is invertible **if and only if** it can be expressed as a product of elementary matrices.

---

Do you want further details on any step or clarification on how elementary matrices work? Here are five questions that expand on this:

1. What are the different types of elementary matrices and how do they relate to elementary row operations?
2. Can you show an example of a matrix being expressed as a product of elementary matrices?
3. How can the inverse of an elementary matrix be computed?
4. What is the role of Gaussian elimination in proving a matrix's invertibility?
5. How does the concept of determinant relate to the invertibility of a matrix?

**Tip:** When proving invertibility, always remember that a matrix is invertible if its determinant is non-zero. This gives a quick criterion for checking invertibility in practice.

Prove that an n × n matrix A is invertible if and only if it can be expressed as a product of elementary matrices.

This proof demonstrates that an n × n matrix is invertible if and only if it can be expressed as a product of elementary matrices. Explore key concepts of linear algebra, including matrix inverses and elementary matrices, with step-by-step explanations.

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