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Invertible matrices

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https://texercises.com/exercise/invertible-matrices/

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Exercise:
Let Ain M_ntimes nK. Place A together with I_n in the following way A|I_n. Then A is invertible iff A|I_n can be brought via a sequence of elementary row-operations to the form I_n|B for some matrix Bin M_ntimes nK. In this case we have A^-B.

Solution:
bf Lemma. Let BCDin M_ntimes nK. Then B C|DB C|B D. Proof of Lemma. Follows directly from the definition of matrix multiplication. Proof of the theorem. Ase A is invertible. By the previous theorem exists elementary matrices T_...T_k s.t. T_k T_k- ... T_ AI_n. Longrightarrow T_k T_k- ... T_ A|I_nI_n|T_k T_k- ... T_. We also have: A B T_^-... T_k-^- T_k^- T_k T_k-... T_...I_n B A T_k^- T_k-^- ... T_ T_^-... T_k-^- T_k^-...I_n &Longrightarrow A^-B Conversely ase that A|I_n can be brought to I_n|B by a sequence of elementary row-operations for some matrix Bin M_ntimes nK Longrightarrow A itself is row-equivalent to I_n Longrightarrow A has rank n Longrightarrow A is invertible.

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Tags	eth, gauss, hs22, lineare algebra, matrices, proof
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Difficulty	(3, default)
Points	0 (default)
Language	(English)
Type	Proof
Creator	rk
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