matrix-notes2 - OPMT 5701 Lecture Notes 1 Matrix Algebra 1....

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OPMT 5701 Lecture Notes 1 Matrix Algebra 1. Gives us a shorthand way of writing a large system of equations. 2. Allows us to test for the existance of solutions to simultaneous systems. 3. Allows us to solve a simultaneous system. DRAWBACK: Only works for linear systems. However, we can often covert non-linear to linear systems. Example y = ax b ln y =ln a + b ln x Matrices and Vectors Given y =10 x x + y y =2+3 x ⇒− 3 x + y =2 In matrix form · 11 31 ¸· x y ¸ = · 10 2 ¸ Matrix of Coe cients Vector of Unknows Vector of Constants In general a 11 x 1 + a 12 x 2 + ... + a 1 n x n = d 1 a 21 x 1 + a 22 x 2 + + a 2 n x n = d 2 .............................. a m 1 x 1 + a m 2 x 2 + + a mn x n = d m n-unknowns ( x 1 , x 2 , ...x n ) Matrix form a 11 a 12 ... a 1 n a 21 a 22 2 n . . . . . . . . . a m 1 a m 2 mn x 1 x 2 . . . x n = d 1 d 2 . . . d m Matrix shorthand Ax = d Where: A= coe cient martrix or an array x= vector of unknowns or an array d= vector of constants or an array Subscript notation a ij is the coe cient found in the i-th row (i=1,. . .,m) and the j-th column (j=1,. . .,n) 1
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1.1 Vectors as special matrices The number of rows and the number of columns de f ne the DIMENSION of a matrix. A is m rows and n is columns or ”mxn.” A matrix containing 1 column is called a ”column VECTOR” xisan × 1 column vector disam × 1co lumnvector If x were arranged in a horizontal array we would have a row vector. Row vectors are denoted by a prime x 0 =[ x 1 ,x 2 ,...,x n ] A1 × 1 vector is known as a scalar. x =[4] is a scalar Matrix Operators If we have two matrices, A and B, then A = Bi f fa ij = b ij Addition and Subtraction of Matrices Suppose A is an m × nmatr ixandBisap × q matrix then A and B is possible only if m=p and n=q. Matrices must have the same dimensions. · a 11 a 12 a 21 a 22 ¸ + · b 11 b 12 b 21 b 22 ¸ = · ( a 11 + b 11 )( a 12 + b 12 ) ( a 21 + b 21 a 22 + b 22 ) ¸ = · c 11 c 12 c 21 c 22 ¸ Subtraction is identical to addition · 94 31 ¸ · 72 16 ¸ = · (9 7) (4 2) (3 1) (1 6) ¸ = · 22 2 5 ¸ Scalar Multiplication Suppose we want to multiply a matrix by a scalar k × A 1 × 1 m × n We multiply every element in A by the scalar k kA = ka 11 12 ... ka 1 n 21 22 2 n . . . m 1 m 2 mn Example Let k= [3] and A= · 62 45 ¸ then kA= = · 3 × 63 × 2 3 × 43 × 5 ¸ = · 18 6 12 15 ¸ Multiplication of Matrices To multiply two matrices, A and B, together it must be true that for A × B = C m × nn × qm × q That A must have the same number of columns (n) as B has rows (n). The product matrix, C, will have the same number of rows as A and the same number of columns as B. 2
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Example A × B = C (1 × 3) (3 × 4) (1 × 4) 1 row 3 rows 1 row 3 cols 4 cols 4 cols In general A × B × C × D = E (3 × 2) (2 × 5) (5 × 4) (4 × 1) (3 × 1) To multiply two matrices: (1) Multiply each element in a given row by each element in a given column (2) Sum up their products Example 1 · a 11 a 12 a 21 a 22 ¸ × · b 11 b 12 b 21 b 22 ¸ = · c 11 c 12 c 21 c 22 ¸ Where: c 11 = a 11 b 11 + a 12 b 21 (sum of row 1 times column 1) c 12 = a 11 b 12 + a 12 b 22
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This note was uploaded on 03/03/2012 for the course ECON 345 taught by Professor Wendywu during the Fall '11 term at Wilfred Laurier University .

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matrix-notes2 - OPMT 5701 Lecture Notes 1 Matrix Algebra 1....

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