Lecture # 4 - System of Linear Equations (cont.) In our last lecture, we were starting to apply the Gaussian Elimination Method to our macro model Y = C + 1500 C = 200 + 4 (Y T ) 5 1 T = 100 + 5 Y We de…ned the extended coe¢ cient matrix 1 1 0 1000 [A d] = 2 4 1 4 200 3 5 5 6 7 6 7 6 1 0 1 100 7 6 5 7 4 5 The objective is to use elementary row operations (ERO’s)to transform our system of linear equations into another, simpler system. – Eliminate coe¢ cient for …rst variable (column) from all equations (rows) except …rst equation (row). – Eliminate coe¢ cient for second variable (column) from all equations (rows) except second equation (rows). – Eliminate coe¢ cient for third variable (column) from all equations (rows) except third equation (rows). The objective is to get a system that looks like this: 1 0 0 s1 0 1 0 s2 2 3 0 0 1 s3 4 5 1 Let’suse our example 1 1 0 1500 [A d] = 2 4 1 4 200 3 5 5 6 7 6 7 6 1 0 1 100 7 6 5 7 4 5 Multiply …rst row (equation) by 1 and add it to third row 5 1 1 0 1500 [A d] = 2 4 1 4 200 3 5 5 6 7 6 7 6 0 1 1 400 7 6 5 7 4 5 Multiply …rst row by 4 and add it to row 2 5 1 1 0 1500 [A d] = 2 0 1 4 1400 3 5 5 6 7 6 7 6 0 1 1 400 7 6 5 7 4 5 Add row 2 to row 3 1 1 0 1500 [A d] = 2 0 1 4 1400 3 5 5 6 7 6 7 6 0 0 9 1800 7 6 5 7 4 5 Multiply second row by 5 1 1 0 1500 [A d] = 2 0 1 4 7000 3 6 7 6 7 6 0 0 9 1800 7 6 5 7 4 5 Add row 2 to row 1 1 0 4 8500 [A d] = 2 0 1 4 7000 3 6 7 6 7 6 0 0 9 1800 7 6 5 7 4 5 2 Multiply row 3 by 5 9 1 0 4 8500 [A d] = 2 0 1 4 7000 3 6 7 6 7 6 0 0 1 1000 7 6 7 4 5 Multiply row 3 by 4 and substract it from row 1 1 0 0 4500 [A d] = 2 0 1 4 7000 3 6 7 6 7 6 0 0 1 1000 7 6 7 4 5 Multiply row 3 by 4 and substract it from row 2 1 0 0 4500 [A d] = 2 0 1 0 3000 3 6 7 6 7 6 0 0 1 1000 7 6 7 4 5 And we get the solution!: Y = 4500;C = 3000;T = 1000 3 In general, there are three possible outcomes: 1 0 0 s1 – 0 1 0 s2 2 3 0 0 1 s3 =4 System has5 a unique solution: ) 1 0 0 s1 – 0 1 0 s2 2 0 0 0 0 3 =4 System has5 in…nitely many solutions ) 1 0 0 s1 – 0 1 0 s2 2 3 0 0 0 s3 =4 System is inconsistent/has5 no solution ) 4 Second method: Find the Inverse Solution for a system of linear equations x = A 1d ) – Before attempting to …nd the inverse: determine whether A is nonsingular Conditions for Nonsingularity of a Matrix 1. Square matrix , i.e., n. equations = n. unknowns. 2. Rows (and columns) linearly independent Best way to see if a matrix is nonsingular: …nd its determinant – If determinant is nonzero matrix is nonsingular ! 5 Determinant De…nition: The determinant is a function that associates a scalar to every square matrix of dimension n Notation: det (A) or A j j Geometric interpretation: the scale factor for volume when the rows of the matrix are regarded as vectors a a For 2x2 matrix A = 11 12 ; the determinant is obtained by multiplying the elements a21 a22 of main diagonal, and substracting the product of the elements of the other diagonal: A = a11 a22 a12 a21 j j Examples 4 3 – If A = ; then A = (4) (2) (3) ( 1) = 11 1 2 j j 3 5 – If B = ; then A = (3) ( 2) (5) (2) = 16 2 2 j j 4 6 – If C = ; then A = (4) (3) ( 2) ( 6) = 0 Singular 2 3 j j ! a11 a12 a13 For 3x3 matrix A = a21 a22 a23 ; there are two ways to …nd determinant: 2 3 a31 a32 a33 4 5 – Sarrus’rule – Laplace Expansion 6 Sarrus’srule: – Write down the determinant twice, except the last column a11 a12 a13 a11 a12 a21 a22 a23 a21 a22 a31 a32 a33 a31 a32 – Multiply the elements of each of the three diagonals from top-left to bottom-right:, and add them: a11a22a33 + a12a23a31 + a13a21a32 = x – Multiply the elements of each of the three diagonals from top-left to bottom-right:, and add them: a13a22a31 + a11a23a32 + a12a21a33 = y – det(A) = A = x y j j 1 1 0 – Example: If A = 1 3 2 ; then A = 2 2 1 0 0 3 j j 4 1 1 0 5 – Example: If B = 1 3 2 ; then B = 2 2 1 2 1 3 j j – Cannot be generalized4 to determinants5 of larger order. 7 Laplace Expansion – Also works for any order of determinant – Need additional de…nitions: Minor: Notation: Mij It is a determinant of the matrix obtained by deleting row i and column j. Cofactor: i+j Notation: Cij = ( 1) Mij It is a minor with a prescribed algebraic sign attached to it Give example for matrix A; B above – Then, the determinant of a 3x3 matrix can be written as A = a11C11+a12C12+a13C13 j j I.e., as the product of a …rst-row element and its corresponding cofactor In this case, we picked row 1, but we can pick any row. 3 – In general A = aijCij; where we picked row i j j j=1 X We can also pick a column n – General rule to …nd a determinant of order n: A = aijCij j j j=1 X 8.