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Unformatted text preview: 1 Lecture Notes 2 1. Review of Matrices The ability to manipulate matrices is critical in economics. 1. Matrix – a rectangular array of numbers, parameters, or variables placed in rows and columns. Matrices are associated with linear equations. Elements ij a denotes the element in row i and column j. column vector – one column of elements row vector – one row of elements Examples 5 1 7 1 8 3 7 6 7 4 3 5 5 2 3 4 3 2 1 8 6 2 3 6 5 4 7 8 9 D C B A (a) What is the dimension of each matrix? Write as G m,n . (b) Which matrix is a square matrix? Column vector? Row vector? (c) What is the value for the following elements a 2,3 , b 1,1 , c 2,1 , and d 3,3 ? 2. Multiplication by a scalar – multiplying every element of a matrix by the scalar Scalar is a number If w = 3, then wA = ? 2 3. Two matrices can be added or subtracted only if they have the same dimensions. The commutative law of addition hold for matrices, H + K = K + H 4 10 4 1 21 2 6 5 4 7 8 9 E A Find A – 2E? 4. matrix multiplication – requires a conformability condition. 3 5 7 8 6 2 4 1 R and H We want to multiply S = H R Check the dimensions for H 3,2 and R 2,1 . The number of columns in H equals the number of rows in R Therefore, the matrices are conformable for multiplication. The dimension of S 3,1 . 61 28 17 3 7 5 8 3 6 5 2 3 4 5 1 3 5 7 8 6 2 4 1 R H S Note – The commutative law of multiplication never holds for matrix multiplication, A B B A The distributive law holds for matrix multiplication, A(B+C)=AB+AC or (B+C)A=BA+BC? 5. An identity matrix (usually denoted by I) is a square matrix with ones in its principle diagonal (the diagonal running northwest to southeast) and zeros everywhere else. 3 Write identity matrices of dimensions: 3 x 3, 5 x 5, and 3 x 6. Any matrix multiplied by the identity matrix gets that same matrix again 7 8 6 2 4 1 1 7 8 1 6 2 1 4 1 7 1 8 6 1 2 4 1 1 1 1 7 8 6 2 4 1 I H 6. Transpose – interchange the rows and columns of a matrix....
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This note was uploaded on 04/05/2012 for the course ECON 421 taught by Professor Blair during the Fall '11 term at Rutgers.
 Fall '11
 BLAIR
 Economics

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