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Matrix Algebra Linear Models A system of m linear equations in n variables: a11 x1 a12 x 2 ... a1n x n d1 a11 a a 21 x1 a 22 x 2 ... a 2 n x n d 2 21 ... .... a m1 x1 a m 2 x 2 ... a mn x n d m am1 a12 a22 ... am 2 ... a1n x1 d1 ... a2 n x2 d 2 Ax d ... ... ... ... ... amn xn d m where a11 a A 21 ... a m1 a12 a 22 ... am2 ... a1n ... a 2 n ... ... ... a mn x1 x x 2 ... xn Each of A, x, d constitutes a matrix. Dimension of the Matrix m × n matrix ( m rows and n columns) a11 a 21 ... am1 a12 a22 ... am 2 ... a1n ... a2 n ... ... ... amn or aij mn Special matrices: 1. 2. m = n n × n square matrix n × 1 column vector x1 x x 2 ... xn n1 3. 1 × n row vector x x1 x2 ... xn 1n 1 d1 d d 2 ... d m Matrix Operations 1. Equality aij bij aij bij mn mn 2. Addition aij bij cij where aij bij cij mn mn mn 3. Subtraction aij bij dij where aij bij d ij mn mn mn 4. Scalar Multiplication k aij 5. mn kaij mn Multiplication of Matrices aik mn bkj n p cij mn 3. 4. In general, AB BA ( AB)C A( BC ) ABC (Associative law of multiplication) 5. A( B C) AB AC; ( B C) A BA CA (Distributive Law) Identity Matrices I 0 ... 0 1 ... 0 (Square matrix) ... ... ... 0 ... 1 nn IA AI A Null Matrices 0 0 ... 0 k 1 Laws 1. A B B A (Commutative law of addition) 2. ( A B) C A ( B C ) A B C (Associative law of addition) 1 0 In ... 0 n where cij aik bkj 0 ... 0 0 ... 0 A A ; A ; A ... ... ... 0 ... 0 mn AB does NOT imply A or B 2 Transpose A or AT A aij 1. 2. 3. mn A a ji nm ( A) A ( A B) A B ( AB) BA Inverses A1 AA1 A1 A I 1. A1 is defined only if A is a square matrix 2. Not every square matrix has an inverse. If A has an inverse, A is said to be nonsingular. If A does not have an inverse, A is said to be singular. 3. 4. 5. A and A1 are inverses of each other. If A is n × n, A1 must be n × n If an inverse exists, it is unique. Evaluating a Determinant 1. First-order determinant a11 a11 2. 3. 4. Second-order determinant a11 a12 a 21 a 22 a11a 22 a12 a 21 Third-order determinant a11 a12 a13 a21 a22 a23 a11a22 a33 a12 a23a31 a21a32 a13 a13a22 a31 a12 a21a33 a23a32 a11 a31 a33 a32 nth-order determinant Minor M ij is obtained by deleting i row and j column. Cofactor Cij (-1)i j M ij n A aij Cij Expansion by the ith row j 1 n A aij Cij Expansion by the jth column i 1 3 Finding the Inverse Matrix A1 C11 C12 adjA C C1n 1 adjA A C 21 C 22 C2n C n1 ... C n 2 ... C nn ... Cramer’s Rule Ax d x j Aj Aj A a11 a12 ... a21 a22 ... d 2 an1 d1 ... a1n ... a2 n an 2 d n ann Exercise 4.6 0 4 3 8 1 0 9 1. Given A ,B and C , find A , B and C . 1 3 0 1 6 1 1 2. Use the matrices given in Prob. 1 to verify that (a) 5.2 A B A B (a) 4 0 1 6 0 3 a b c (e) b c a c a b 4. Test whether the following matrices are nonsingular: 4 0 1 (a) 19 1 3 7 1 0 5.4 AC C A 1. Evaluate the following determinants: 8 1 3 5.3 (b) 4 9 5 (d) 3 0 1 10 8 6 4. Find the inverse of each of the following matrices: 4 2 1 (a) E 7 3 0 2 0 1 1 0 0 (c) G 0 0 1 0 1 0 4 5.5 3. Use Cramer’s rule to solve the following equation system: 8 x1 x 2 16 (a) 2 x 2 5 x3 5 2 x1 3 x3 7 x y z a (d) x y z b x y z c 5 Differentiation The Definition of Derivative f x dy f x x f x lim dx x0 x Rules of Differentiation 1. 2. 3. 4. dc 0 dx (c: constant; Constant-function Rule) dx n nx n 1 (Power-function Rule) dx d cf ( x) c d f ( x) cf ( x) dx dx d f ( x) g ( x) d f ( x) d g ( x) f ( x) g ( x) (Sum-difference dx dx dx Rule) 5. 6. 7. 8. d f ( x) g ( x) f ( x) g ( x) f ( x) g ( x) (Product Rule) dx d f ( x) f ( x) g ( x) f ( x) g ( x) (Quotient Rule) dx g ( x) g 2 ( x) dx 1 dy dy dx (Inverse-function Rule) z f ( y ); y g ( x) dz dz dy dx dy dx (Chain Rule) The Definition of Partial Derivative f x1 x1 , x2 ,..., xn f x1 , x2 ,..., xn y f1 lim x1 x0 x1 Total Differentials U U ( x1 , x 2 ,..., x n ) dU 1. dc 0 2. 3. du n nu n 1 du d (u v) du dv 4. U U U dx1 dx 2 ... dx n x1 x 2 x n d uv vdu udv 6 5. 6. 7. u 1 d 2 vdu udv v v d (u v w) du dv dw d uvw vwdu uwdv uvdw Total Derivatives Q QK t , Lt , t Q Q Q dK dL dt K L t dQ Q dK Q dL Q dt K dt L dt t dQ Derivatives of Implicit Functions Explicit function: y f ( x1 , x2 ,..., xn ) Implicit function: F ( x1 , x2 ,..., xn ) 0 F ( x, y ) 0 dF F F F dy dx dy Fx dx Fy dy 0 x x y dx Fy 7 Exponential Functions Simple Exponential Function y f (t ) b t Graphical Form y (b>0; y>0) y = 2t 2 1 t 0 A Preferred Base e lim (1 m 1 y = et 1 m ) 2.71828... m Logarithmic Functions The Meaning of Logarithm Log functions are inverse functions of certain exponential functions. y b t t log b y (b>0; y>0) Common Log and Natural Log 10 is the base Common Log (e.g. log10100 = log100 = 2) e is the base Natural Log (e.g. loge100 = ln100 = 4.60517…) Rules of Logarithm Rule 1: logc(ab) = logc(a) + logc(b) Rule 2: logc(a/b) = logc(a) – logc(b) Rule 3: logc (ba) =a logc(b) Rule 4: logc (a) = logc (b) logb (a) Rule 5: logc (a) = 1/ logc (a) The Graphical Form 8 y y y = et t = ln y 1 45˚ t 0 45˚ 0 1 Log-Function Rule d 1 ln t dt t Exponential-Function Rule d t e et dt The Rules Generalized d f (t ) e f ' (t )e f (t ) dt d f ' (t ) ln f (t ) dt f (t ) t The Case of Base b d t b b t ln b dt d 1 log b t dt t ln b More general formulas: d f (t ) b f ' (t )b f (t ) ln b dt d f ' (t ) 1 log b f (t ) dt f (t ) ln b 9 Exercise 7.1 2. Find the following: (a) d x 4 dx d au b du (f) 7.2 3. Differentiate the following by using the product rule: (b) 3x 10 6 x 2 7 x (e) 2 3x1 xx 2 8.3 2. Use the rules of differentials to find dy from the following functions x1 2 x1 x 2 (a) y (b) y x1 x 2 x1 x 2 Given y 3x1 2 x2 1x3 5 3. (a) Find dy by rule VII (b) Find the differential of y, if dx2 dx3 0 . 8.4 1. Find the total derivative dz/dy, given (a) z f x, y 5x xy y 2 , where x g y 3y 2 10.5 1. Find the derivatives of: bx c (e) y e ax 3. (g) 2x y ln 1 x 4. (b) y log 2 t 1 2 (f) y xe x (c) y 132t 3 10 (f) y x 2 log 3 x Integration Indefinite Integrals d F x f x f x dx F x c dx Rules of Integration 1. x n 1 x dx n 1 c n 1 2. e dx e 3. 1 xdx ln x c x 0 the logarithmic rule 4. f x g x dx f x dx g x dx the integral of a sum 5. k f x dx k f x dx the integral of a multiple 6. du f u x dx dx f u du F u c the substitution rule 7. vdu uv udv integration by parts x x c the exponential rule Definite Integrals b a the power rule n f x dx F x a F b F a b Properties of Definite Integrals a f x dx f x dx b 1. 2. f x dx F a F a 0 3. 4. 5. b a a a d b b a a a f x dx f x dx f x dx f x dx a b c d b c d a b c f x dx f x dx b a kf x dx k f x dx b a 11 Improper Integrals - Infinite Limits of Integration b f x dx lim f x dx; b b a a f x dx f x dx lim f x dx b a a - f x dx lim b a a b Infinite Integrand Assume that f x as x p , where p is a point in the interval (a, b); then f x dx f x dx f x dx b p a b a p The given integral on the left can be considered as convergent if and only if each subintegral has a limit. Exercise 14.2 1. Find the following: (d) 2. 1. (e) dx x 4x dx 1 (e) 2 2ax b ax 3e 2 x 7 (e) dx 4 xe x2 3 dx Find: (a) 14.3 2 x Find: (d) 4. 2e x 3 x 1 12 (b) dx x ln xdx Evaluate the following: (c) 3 1 3 xdx (e) ax 1 1 2 bx c dx 2. Evaluate the following: 14.4 3. 1 1 dx e x 1 x Evaluate the following: (d) (c) 6 1 0 x2 3dx (d) 0 ert dt (e) 12 5 1 dx x2 2 7 bx dx Optimization Problems One variable First derivative versus Second derivative f ( x) f ( x) f ( x) f x0 0 f x at x0 f x0 0 f x f x0 0 f x at x0 f x0 0 f x Relative (local) versus Absolute (global) Extremum 1. Absolute (global) extremum Relative (local) Extremum First-derivative test for relative extremum If f x0 0 1. 2. 3. A relative minimum if f x change its sign - + from the immediate left of x 0 to its immediate right. Neither a relative maximum nor a relative minimum if f x has the same sign on both the immediate left and immediate right of x 0 Second-derivative test for relative extremum If f x0 0 1. A relative maximum if f x0 0 . 2. A relative minimum if f x0 0 3. A relative maximum if f x change its sign + - from the immediate left of x 0 to its immediate right. Either a relative maximum, or a relative minimum, or an inflection point if f x0 0 . Conditions for a relative extremum y f (x) Condition First-order condition Second-order condition Maximum f x 0 f x 0 Minimum f x 0 f x 0 13 Nth-derivative test for relative extremum If f x0 0 and if the first nonzero derivative is that of Nth derivative, f (N) x0 0 . 2. A relative maximum if f ( N ) x 0 and N is an even number. A relative minimum if f ( N ) x 0 and N is an even number. 3. An inflection point if N is odd. 1. More than one variable Conditions for a relative extremum z f ( x, y ) Condition Maximum fx fy 0 First-order condition f xx 0; Second-order condition Minimum f fx fy 0 yy 0 ; f xx 0; f xx f yy f xy2 Hessian (determinant) f yy 0 ; f xx f yy f xy2 H A Hessian is a determinant with the second-order partial derivatives as its elements. 1. In the two-variable case, H f xx f xy f yx f yy f xx f yy f xy f yx f xx f yy f xy 2 Note that f xy f yx (Young’s theorem) 2. In the n-variable case, H f 11 f 12 ... f 1n f 21 f 22 ... f 2n ... ... ... ... f n1 f n2 ... f nn H1 f11 ; H 2 ; f11 f12 f 21 f 22 f 11 f 12 f 13 ; H 3 f 21 f 31 f 22 f 23 ; … … H n H f 33 14 f 32 Determinantal Test for a relative extremum z f ( x1 , x2 ,..., xn ) Condition First-order condition Second-order condition Maximum f1 f 2 ... f n 0 Minimum f1 f 2 ... f n 0 H 2 0; H 3 0... H 2 , H 3 ,..., H n 0 1n H n 0 Optimization with Equity Constraints Free optimum versus constrained optimum Lagrange-Multiplier Method z f ( x1 , x2 ,..., xn ) subject to the constrain g ( x1 , x2 ,..., xn ) c Lagrangian function Z f ( x1, x 2 ,..., x n ) c g ( x1 , x 2 ,..., x n ) First-order condition Z c g ( x1 , x 2 ,..., x n ) 0 Z 1 f 1 g 1 0 Z 2 f 2 g 2 0 ........................ Z n f n g n 0 Bordered Hessian H 0 g1 g2 ... g1 Z 11 Z 12 ... Z 1n H g2 Z 21 Z 22 ... Z 2 n ... ... ... gn Z n1 Z n2 0 g1 g2 H 2 g1 Z 11 g2 Z 21 ... gn ... ... Z nn 0 g1 g2 g3 g Z 12 ; H 3 1 g2 Z 22 g3 Z 11 Z 12 Z 13 Z 21 Z 22 Z 23 Z 31 Z 32 Z 33 15 ;….. H n H Determinantal Test for a relative extremum z f ( x1 , x2 ,..., xn ) subject to g ( x1 , x2 ,..., xn ) c ; with Z f ( x1, x 2 ,..., x n ) c g ( x1 , x 2 ,..., x n ) Condition First-order condition Second-order condition Maximum Z Z1 Z 2 ... Z n 0 H 2 0; H 3 0; H 4 0... 1n H n Minimum Z Z1 Z 2 ... Z n 0 H 2 , H 3 ,..., H n 0 0 Exercise 11.4 Find the extreme values, if any, of the following four functions. Check whether they are maxima or minima by the determinantal test. 1. z x12 3x 22 3x1 x 2 4 x 2 x3 6 x32 2. z 29 x12 x 22 x32 3. z x1 x3 x12 x 2 x 2 x3 x 22 3x32 12.2 1. Use the Lagrange-multiplier method to find the stationary value of z: (a) z xy , subject to x 2 y 2 (b) z x y 4 , subject to x y 8 (c) z x 3 y xy , subject to x y 6 (d) z 7 y x 2 , subject to x y 0 16