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slide2-aug24 - ECON 401 Matrices Siyang Xiong Rice...

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ECON 401: Matrices Siyang Xiong Rice University August 23, 2011 Xiong (Rice University) ECON 401 August 23, 2011 1 / 24
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Matrice n equations of m variable a 11 x 1 C a 12 x 2 C ... C a 1 m x m D b 1 (1) a 21 x 1 C a 22 x 2 C ... C a 2 m x m D b 2 ... a n 1 x 1 C a n 2 x 2 C ... C a nm x m D b n (1) can be represented as follows: 2 4 a 11 ... a 1 m ... ... ... a n 1 ... a nm 3 5 2 4 x 1 ... x m 3 5 D 2 4 b 1 ... b n 3 5 Xiong (Rice University) ECON 401 August 23, 2011 2 / 24
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matrice consider a n ° m matrice A , A D 2 4 a 11 ... a 1 m ... ... ... a n 1 ... a nm 3 5 , for example b A D ° 1 2 4 2 0 2 ± row vectors, denoted as A r i , are de°ned as: b A r i D ² a i 1 ... a im ³ ; b A r 1 D ² 1 2 4 ³ ; b A r 2 D ² 2 0 2 ³ . sometimes we write, A D 2 4 A r 1 ... A r n 3 5 . Xiong (Rice University) ECON 401 August 23, 2011 3 / 24
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matrice consider a n ° m matrice A , A D 2 4 a 11 ... a 1 m ... ... ... a n 1 ... a nm 3 5 , for example b A D ° 1 2 4 2 0 2 ± column vectors, denoted as A c j , are de°ned as: A c j D 2 4 a 1 j ... a nj 3 5 ; b A c 1 D ° 1 2 ± ; b A c 2 D ° 2 0 ± ; b A c 3 D ° 4 2 ± . sometimes we write, A D ² A c 1 ... A c m ³ Xiong (Rice University) ECON 401 August 23, 2011 4 / 24
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addition of matrices given two n ° m matrices A and B , A C B D 2 4 a 11 C b 11 ... a 1 m C b 1 m ... ... ... a n 1 C b n 1 ... a nm C b nm 3 5 for example: A D ° 1 2 4 2 0 2 ± and B D ° 3 2 1 0 4 2 ± , A C B D ° 1 C 3 2 C 2 4 C 1 2 C 0 0 C 4 2 C 2 ± D ° 4 4 5 2 4 4 ± Xiong (Rice University) ECON 401 August 23, 2011 5 / 24
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multiplication of matrices given a n ° m matrice A and a m ° k matrice B , AB D 2 6 6 6 6 4 A r 1 ± B c 1 ... A r 1 ± B c k ... ... ... ... A r i ± B c j ... ... ... ... A r n ± B c 1 ... A r n ± B c k 3 7 7 7 7 5 is a n ° k matrice, D 2 6 6 6 6 4 P m h D 1 A 1 h ± B h 1 ... P m h D 1 A 1 h ± B hk ... ... ... ... P m h D 1 A ih ± B hj ... ... ... ... P m h D 1 A nh ± B h 1 ... P m h D 1 A nh ± B hk 3 7 7 7 7 5 . Xiong (Rice University) ECON 401 August 23, 2011 6 / 24
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multiplication of matrices for example: A D 2 4 1 ² 1 2 0 1 3 3 5 and B D ° 1 2 4 2 0 2 ± : AB D 2 4 1 ² 1 2 0 1 3 3 5 ° 1 2 4 2 0 2 ± D 2 4 1 ° 1 C .
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