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Chapter 2 : Matrices

Chapter 2 : Matrices. 2.1 Operations with Matrices (Study Book 2.1 & Larson 2.1). Matrices are called equal if & only if they have the same size AND corresponding entries are equal. A = 2 -1 5 is not equal to 2 -1 5

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Chapter 2 : Matrices

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  1. Chapter 2 : Matrices 2.1 Operations with Matrices (Study Book 2.1 & Larson 2.1)

  2. Matrices are called equal if & only if • they have the same size AND • corresponding entries are equal. A = 2 -1 5 is not equal to 2 -1 5 1 9 3 1 9 3 0 0 0 Matrices of the same size can be added by adding corresponding entries: ie the i,j th entry of A + B is aij + bij Examples:

  3. Scalar multiplication is also defined entry-wise: Eg: if A = 2 -1 5 1 9 3 then 3A = 6 -3 15 3 27 9 and - 1A = -2 1 -5 = -A -1 -9 -3

  4. But Matrix Multiplication is not just the multiplication of corresponding entries! A more useful definition of AB has the i,j entry formed by taking the scalar product of the ith row of A with the jth column of B. Example: 2 -1 5 0 1 -1 = 9 4 3 1 9 3 1 -2 0 15 -17 2 2 0 1 More…

  5. AB is defined only if A is m x n , B is n x p. • ie their inner dimensions must agree. • AB then has size m x p. • The ij th entry of the product • a11 a12 a13 … a1 nb11 b12 ... b1 j … b1p • a21 a22 a23 … a2 n b21 b22 … b2 j … b2 p • … … … ... b31 b32 … b3 j … b3p • ai 1 ai 2 ai 3 … ai n ... … … ... • … … … ... bn1 bn2 … bn j… bnp • am1 am2 am3 … am n • is the dot product ( i th row of A ) . ( j th col of B ) • ie c i j = ai1 b1j + ai2 b2 j + ai 3 b3 j + … + ainbnj .

  6. One reason for this unusual defn of multiplication is that it can be used to write a whole system of linear equationsas just 1 matrix equation. Eg: The system 2 x + 3 y - 5 z = 8 - x + 5 y + 7z = 3 6x + 2y = 1 7x - 4y + z = 3 can be written as a matrix product as below: 2 3 -5 x = 8 -1 5 7 y 3 6 2 0 z 1 7 -4 1 3 Note Inner dimensions agree! (4 x 3) (3 x 1) = (4 x 1)

  7. Hence the system has matrix form A x = b, where • A is the matrix of coefficients • x the column of unknowns • b the column of RHS’s (or “demands”) In reverse, given an equation of form Ax = b, multiplying out the product on the LHS gives a system of linear equations. Eg: Write out the equations represented by 2 -1 5 x = -2 1 9 3 y 6 z

  8. Homework Larson & Edwards Ch 2.1: • Master the odd numbers 5 - 31. • Also do Ed 4: Q 37, 50, 51, 52; or Ed 5: Q 39, 54, 55, 56. • Write full solutions to Ed 4: Q 7, 15, 23, 25, 41, 50, 51; or Ed 5: Q 7, 15, 23, 25, 43, 54, 55.

  9. Objectives Be able to • add matrices • find a scalar multiple of a matrix • multiply matrices • manipulate matrix equations • write a system of linear equations as a matrix equation

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