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3.1 Linear Equations

# 3.1 Linear Equations - 3.1 3 Elementary Matrix Operations...

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3.1. § 3. Elementary Matrix Operations and Systems of Linear Equations 3.1. Elementary Matrix Operations and Elementary Ma- trices 1. (a) T (b) F ( ) I 3 ˆ 1 0 - 2 0 1 0 0 0 1 , by - 2 × C 1 + C 3 C 3 (c) T ( ) I n ˆ I n , by 1 × C 1 C 1 (d) F (e) T ( ) Theorem 3.2 (f) F ( ) Let E 1 = 1 0 0 0 - 1 0 0 0 1 , E 2 = 1 0 0 0 1 0 0 0 - 1 then, E 1 + E 2 is not an elementary matrix (g) T (h) F (i) T Let A ˆ B = EA , then E is invertible and its inverse is also an elementary matrix 154 PNU-MATH

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3.1. B ˆ A = E - 1 B 2. (i) A ˆ B , by - 2 × C 1 + C 2 C 2 (ii) B ˆ C , by - 1 × R 1 + R 2 R 2 (iii) C ˆ I 3 , by 1 2 × R 2 R 2 R 2 R 3 R 3 + R 2 R 2 1 4 × R 2 R 2 - 1 × R 2 + R 3 R 3 - 3 × R 3 + R 1 R 1 3. (a) E = 0 0 1 0 1 0 1 0 0 I 3 ( R 1 R 3 ) E ( R 1 R 3 ) I 3 E - 1 = 0 0 1 0 1 0 1 0 0 (b) E = 1 0 0 0 3 0 0 0 1 I 3 (3 × R 2 R 2 ) E ( 1 3 × R 2 R 2 ) I 3 155 PNU-MATH
3.1. E - 1 = 1 0 0 0 1 3 0 0 0 1 (c) E = 1 0 0 0 1 0 - 2 0 1 I 3 ( - 2 × R 1 + R 3 R 3 ) E (2 × R 1 + R 3 R 3 ) I 3 E - 1 = 1 0 0 0 1 0 2 0 1 4.

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3.1 Linear Equations - 3.1 3 Elementary Matrix Operations...

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