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.

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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.