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Unformatted text preview: • Theorem 1.5.1 suggests that reducing a matrix A to (reduced) row echelon form is tha same as multiplying A from left by the appropriate elementary matrices. • Hence if B is a matrix obtained from a matrix A by performing a finite sequence of elementary row operations, say k many, then there exists a sequence of elementary matrices E 1 ,E 2 ,...,E k such that E k E k 1 ··· E 2 E 1 A = B Definition: Matrices A and B are called row equivalent if either (hence each) can be obtained from the other by a sequence of elementary row operations. In our example following Theorem 1.5.1 we have E 2 E 1 A = 1 2 0 1 = B In fact we can apply one more elementary row operation to B and get 1 2 0 1 2 R 2 + R 1 → 1 0 0 1 E 3 : 1 0 0 1 2 R 2 + R 1 → 1 2 1 ⇒ E 3 E 2 E 1 A = 1 0 0 1 = I Hence E 3 E 2 E 1 is a candidate for A 1 . In fact we will see that A 1 = E 3 E 2 E 1 . Theorem 1.5.2: Every elementary matrix is invertible, and the inverse is also an elementary matrix. Observation: I E I cR i 1 c R i inverse of E is the one obtained by applying 1 c R i to I I E I R i ↔ R j R j ↔ R i inverse of E is the one obtained by switching back R i and R j I E I cR i + R j R j cR i inverse of E is the one obtained by substracting c mupltiple of R i from R j Examples: 1. 1 0 0 1 2 R 1 2 0 0 1 1 2 R 1 1 0 0 1 ⇒ 2 0 0 1 1 = 1 2 0 1 1 2. 1 0 0 1 R 1 ↔ R 2 → 0 1 1 0 R 1 ↔ R 2 → 1 0 0 1 ⇒ 0 1 1 0 1 = 0 1 1 0 3. 1 0 0 1 2 R 1 + R 2 → 1 0 2 1 2 R 1 + R 2 → 1 0 0 1 ⇒ 1 0 2 1 1 = 1 2 1 Theorem 1.5.3: If A is an n × n matrix, then the following statements are equivalent, that is, all true or all false. 1. A is invertible. 2. Ax = 0 has only the trivial solution. 3. The reduced row echelon form of A is I n . 4. A is expressible as a product of elementary matrices. Proof: ( a ) ⇒ ( b ) A invertible ⇒ A 1 ( Ax ) = A 1 ⇒ Ix = 0 ⇒ x = 0 ( b ) ⇒ ( c ) let x = x 1 x 2 . . . x n , Ax = 0 has only nontrivial solution means every unknown is a leading variable. This means the reduced row echelon form of the augmented matrix will be 1 0 ......
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This note was uploaded on 04/27/2011 for the course ENGR 130 taught by Professor Zhang during the Spring '11 term at University of Alberta.
 Spring '11
 Zhang

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