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**Unformatted text preview: **Recall that any system of linear equations can be represented as a matrix equation: A x = b Sec 3.5 Saturday, February 06, 2010 11:20 PM Section3dot5 Page 1 Recall that any system of linear equations can be represented as a matrix equation: A x = b If the system has n equations and n unknowns, then A is a square matrix and we can solve the matrix equation using A-1 , if it exists 3.5.1 Saturday, February 06, 2010 11:20 PM Section3dot5 Page 2 Recall that any system of linear equations can be represented as a matrix equation: A x = b If the system has n equations and n unknowns, then A is a square matrix and we can solve the matrix equation using A-1 , if it exists Equivalently, if det( A ) 0, we can solve for x : x = A-1 b 3.5.2 Saturday, February 06, 2010 11:20 PM Section3dot5 Page 3 Recall that any system of linear equations can be represented as a matrix equation: A x = b If the system has n equations and n unknowns, then A is a square matrix and we can solve the matrix equation using A-1 , if it exists Equivalently, if det( A ) 0, we can solve for x : x = A-1 b det( A ) 0 A-1 exists A x = b has a unique solution for any b 3.5.3 Saturday, February 06, 2010 11:20 PM Section3dot5 Page 4 Recall that any system of linear equations can be represented as a matrix equation: A x = b If the system has n equations and n unknowns, then A is a square matrix and we can solve the matrix equation using A-1 , if it exists Equivalently, if det( A ) 0, we can solve for x : x = A-1 b det( A ) 0 A-1 exists A x = b has a unique solution for any b The row rank of A is n REASON: If A-1 exists, then rref( A ) = I n Hence there are n non-zero rows in ref( A ) 3.5.4 (*) Saturday, February 06, 2010 11:20 PM Section3dot5 Page 5 If...

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