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lecture11hout - MATH 006 Calculus and Linear...

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MATH 006 Calculus and Linear Algebra (Lecture 11) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 12
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Outline 1 Matrix Equations 2 An Application Albert Ku (HKUST) MATH 006 2 / 12
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Matrix Equations Matrix Equations Consider the following system x 1 - x 2 + x 3 = 1 2 x 2 - x 3 = 1 2 x 1 + 3 x 2 = 1 We can rewrite the system as a matrix equation . The idea is to define the following matrices: A = 1 - 1 1 0 2 - 1 2 3 0 , x = x 1 x 2 x 3 , b = 1 1 1 Albert Ku (HKUST) MATH 006 3 / 12
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Matrix Equations Then we observe that the system can be rewritten as Ax = b . This is a matrix equation whose unknown is x . To solve a general matrix equation of the form Ax = b , we first assume that A is an invertible square matrix (a square matrix whose inverse exists). Then we have A - 1 Ax = A - 1 b Ix = A - 1 b x = A - 1 b That is to say, if we can find A - 1 , then we can solve the matrix equation and hence the system. Albert Ku (HKUST) MATH 006 4 / 12
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Matrix Equations Finding Inverse of A 1 - 1 1 1 0 0 0 2 - 1 0 1 0 2 3 0 0 0 1 R 3 +( - 2) R 1 R 3 -→ 1 - 1 1 1 0 0 0 2 - 1 0 1 0 0 5
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