# 1.6 - MATH1850U/2050U Chapter 1 cont 1 LINEAR SYSTEMS cont...

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MATH1850U/2050U: Chapter 1 cont. .. 1 LINEAR SYSTEMS cont… More on Linear Systems and Invertible Matrices (1.6; pg. 60) Recall: Last day, we learned about the concept of inverting a matrix. Theorem: Every system of linear equations has either no solution, exactly one solution, or infinitely many solutions. Proof (that if more than one solution, then infinitely many): Theorem: If A is an invertible n n matrix, then for each column vector b , the system of equations b x A has exactly one solution, namely b x 1 A Proof:

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MATH1850U/2050U: Chapter 1 cont. .. 2 Example: Consider the following system with the information provided. ..what is the solution to the system? Question: Say you wanted to solve ALL the systems , , , 3 2 1 b x b x b x A A A where A is the same for all systems. How could do this efficiently? Idea: If A is invertible, then , , , 3 1 3 2 1 2 1 1 1 b x b x b x A A A or set up the augmented matrix system  | | | | 3
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1.6 - MATH1850U/2050U Chapter 1 cont 1 LINEAR SYSTEMS cont...

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