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Given the matrix $\mathbf A = \begin{pmatrix}3&5\\2&4\end{pmatrix}$, how would I go about writing this as a product of elementary matrices? I understand the concept of elementary matrices I'm just a little unsure algorithmically what the steps should be. Any help would be appreciated.

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Perform the Gauss algorithm to bring $A$ into the reduced echelon form. Because every step of the Gauss algorithm is basically a multiplication with an elementary matrix you get the equation (You can also get the $T_i$ on the left side of $A$ it depends if you do row or column operations)

$$A \cdot T_1 \cdot T_2 \cdot \ldots \cdot T_k = \mathcal{E}$$

Where $\mathcal{E}$ is the unit matrix. Therefore you have that

$$A = T_k^{-1} \cdot \ldots \cdot T_2^{-1} \cdot T_1^{-1}$$

And because the inverse of elementary matrices are elementary matrices you have written $A$ as the product of elementary matrices. This is a constructive algorithm you just have to keep track of the elementary matrices you use.

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