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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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