Chapter 1.4 ElementaryMatrices

3 add 2 times row 1 to row 3 add 2 times row 1 to row

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Unformatted text preview: times row 1 to row 3. 1 Interchange rows 2 and 3. Multiply row 2 by . You end up 3 with I3 . Outline Elementary Matrix Inverse of an Elementary Matrix The Inverse of a Matrix Properties of an Invertible Matrix Theorem If A is an n × n matrix, then the following are equivalent. (a) A is invertible. (b) Ax = 0 has only the trivial solution. (c) rref (A) = In . (d) A = E1 E2 . . . Er for some elementary matrices E1 , E2 , . . . , Er . Proof. (a) ⇒ (b): (b) ⇒ (c): (c) ⇒ (d): (d) ⇒ (a): Outline Elementary Matrix Inverse of an Elementary Matrix The Inverse of a Matrix Row Equivalent Matrices If a matrix B can be obtained from a matrix A by a finite sequence of elementary row operations, then A can be obtained from B with the reverse sequence of the inverse elementary row operations. We say A and B are row equivale...
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