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

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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

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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. Let’s 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 = 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

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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 = x y z , B = 20 - 6 .
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section14 - Chapter 1 MAT188H1F Lec03 Burbulla Chapter 1...

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