section14 - Chapter 1 MAT188H1F Lec03 Burbulla Chapter 1...

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Unformatted text preview: Chapter 1 MAT188H1F Lec03 Burbulla Chapter 1 Lecture Notes Fall 2007 Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 1 Chapter 1 Matrix Multiplicaton Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 1 Matrix Multiplicaton Some Introductory Examples The product ST of the m p matrix S and the q n matrix T is only defined if p = q ; that is, the number of columns of S must match the number of rows of T . Consider the four matrices A = 1- 1 3 2 4- 2 , B = 3 7 2 5 , C = 4- 1 5 , D =- 2 5 9 4 . A is 2 3; B is 2 2; C is 3 1; and D is 2 2 . Only the following products will be defined: BA , AC , DA , BD , and DB . Why? Because the definition of matrix multiplication requires you to match the elements in the rows of one matrix with the elements in the columns of the other. Matrix multiplication is a complicated matter! but it turns out to be extremely useful. Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 1 Matrix Multiplicaton Example 1 Consider the product BA . Here is how we multiply them: BA = 3 7 2 5 1- 1 3 2 4- 2 = 3(1) + 7(2) 3(- 1) + 7(4) 3(3) + 7(- 2) 2(1) + 5(2) 2(- 1) + 5(4) 2(3) + 5(- 2) = 17 25- 5 12 18- 4 Similarly, DA =- 2 5 9 4 1- 1 3 2 4- 2 = 8 22- 16 17 7 19 , where I did the calculations in my head. As should you! Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 1 Matrix Multiplicaton Example 2 Both products BD and DB are possible. Lets calculate them: BD = 3 7 2 5- 2 5 9 4 = 57 43 41 30 . DB =- 2 5 9 4 3 7 2 5 = 4 11 35 83 . Note that BD 6 = DB . Be careful! matrix multiplication has properties different than number multiplication. Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 1 Matrix Multiplicaton Example 3 AC = 1- 1 3 2 4- 2 4- 1 5 = 20- 6 Why is such multiplication useful? Consider AX , if X = x y z : AX = 1- 1 3 2 4- 2 x y z = x- y + 3 z 2 x + 4 y- 2 z Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 1 Matrix Multiplicaton Thus the system of equations x- y + 3 z = 20 2 x + 4 y- 2 z =- 6 can be written in matrix form AX = B , with A = 1- 1 3 2 4- 2 , X =...
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This note was uploaded on 06/22/2008 for the course ENGINEERIN MAT188 taught by Professor Burbula during the Fall '07 term at University of Toronto- Toronto.

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section14 - Chapter 1 MAT188H1F Lec03 Burbulla Chapter 1...

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