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Machina Math Handout

# Have to take into account the direct effect of on the

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Unformatted text preview: solve systems of linear equations we need to define the determinant |A| of a square matrix A. If A is a 1 × 1 matrix, that is, if A = [a11], we define |A| = a11. In the 2 × 2 case: if ⎡a A = ⎢ 11 ⎣ a21 a12 ⎤ a22 ⎥ ⎦ we define |A| = a11 ⋅ a22 − a12 ⋅ a21 that is, the product along the downward sloping diagonal (a11⋅a22), minus the product along the upward sloping diagonal (a12⋅a21). In the 3 × 3 case: if ⎡ a11 A = ⎢ a21 ⎢ ⎢ a31 ⎣ a12 a22 a32 a13 ⎤ a23 ⎥ ⎥ a33 ⎥ ⎦ then first form ⎡ a11 ⎢a ⎢ 21 ⎢ a31 ⎣ a12 a22 a32 a13 ⎤ a11 a23 ⎥ a21 ⎥ a33 ⎥ a31 ⎦ a12 a22 a32 (i.e., recopy the first two columns). Then we define: |A| = a11⋅a22⋅a33 + a12⋅a23⋅a31 + a13⋅a21⋅a32 – a13⋅a22⋅a31 – a11⋅a23⋅a32 – a12⋅a21⋅a33 in other words, add the products of all three downward sloping diagonals and subtract the products of all three upward sloping diagonals. Unfortunately, this technique doesn’t work for 4×4 or bigger matrices, so to hell with them. Systems of Linear Equations and Cramer’s Rule The general form of a system of n linear equations in the n unknown variables x1,..., xn is: a11 ⋅ x1 + a12 ⋅ x2 + + a1n ⋅ xn = c1 a21 ⋅ x1 + a22 ⋅ x2 + + a 2 n ⋅ xn = c2 an1 ⋅ x1 + an 2 ⋅ x2 + + ann ⋅ xn = cn ⎡ a11 a12 ⎢a a22 for some matrix of coefficients A = ⎢ 21 ⎢ ⎢ ⎣ a n1 a n 2 a1n ⎤ a2 n ⎥ ⎥ and vector of constants C = ⎥ ⎥ ann ⎦ ⎡ c1 ⎤ ⎢c ⎥ ⎢ 2⎥ ⎢⎥ ⎢⎥ ⎣ cn ⎦ Note that the first subscript in the coefficient aij refers to its row and the second subscript refers to its column (thus, aij is the coefficient of xj in the i’th equation). We now give Cramer’s Rule for solving linear systems. The solutions to the 2 × 2 linear system: a11 ⋅ x1 + a12 ⋅ x2 a21 ⋅ x1 + a22 ⋅ x2 Econ 100A = c1 = c2 18 Mathematical Handout are simply: * x1 = c1 a12 c2 a22 a11 a12 a21 a22 and * x2 = a11 c1 a21 c2 a11 a12 a21 a22 The solutions to the 3 × 3 system: a11 ⋅ x1 + a12 ⋅ x2 + a13 ⋅ x3 = c1 a21 ⋅ x1 + a22 ⋅ x2 + a23 ⋅ x3 = c2 a31 ⋅ x1 + a32 ⋅ x2 + a33 ⋅ x3 = c3 are simply: ∗ x1 = c1 a12 a13 c2 a22 a23 c3 a32 a33 a11 a12 a13 a21 a22 a23 a31 a32 a33 ∗ x2 = a11 a21 a31 a11 a21 a31 c1 a13 c2 a23 c3 a33 a12 a13 a22 a23 a32 a33 ∗ x3 = a11 a12 c1 a21 a22 c2 a31 a32 c3 a11 a12 a13 a21 a22 a23 a31 a32 a33 Note that in both the 2 × 2 and the 3 × 3 case we have that x* is obtained as the ratio of two i determinants. The denominator is always the determinant of the coefficient matrix A. The numerator is the determinant of a matrix which is just like the coefficient matrix, except that the j’th column has been replaced by the vector of right hand side constants. Econ 100A 19 Mathematical Handout...
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