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https://texercises.com/exercise/expressing-matrices-through-elementary-matrices/
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Exercise:
forall matrices Ain textGL_nK exists kgeq and elementary matrices T_...T_k s.t. T_k T_k-... T_AI_n. Furthermore we have: AT_^-... T_k^- and A^-T_k... T_. In particular A can be written as a finite product of elementary matrices.

Solution:
Proof. We already know that every Ain M_ntimes nK can be brought to a row-reduced echelon matrix A using the Gauss elimination algorithm. In our case A is invertible hence textrankAn.quad textBut textrankA'textrankAquad texthence textrankA'n &Longrightarrow A'I_n. But each step in the Gauss elimination algorithm is an elementary operation hence exists elementary matrices T_...T_k of types I II III s.t. T_k T_k-... T_AI_n. Longrightarrow AT_^-... T_k^-.
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Exercise:
forall matrices Ain textGL_nK exists kgeq and elementary matrices T_...T_k s.t. T_k T_k-... T_AI_n. Furthermore we have: AT_^-... T_k^- and A^-T_k... T_. In particular A can be written as a finite product of elementary matrices.

Solution:
Proof. We already know that every Ain M_ntimes nK can be brought to a row-reduced echelon matrix A using the Gauss elimination algorithm. In our case A is invertible hence textrankAn.quad textBut textrankA'textrankAquad texthence textrankA'n &Longrightarrow A'I_n. But each step in the Gauss elimination algorithm is an elementary operation hence exists elementary matrices T_...T_k of types I II III s.t. T_k T_k-... T_AI_n. Longrightarrow AT_^-... T_k^-.
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eth, hs22, lineare algebra, matrices, proof
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