Question

3. There are three types of life insurance policies that a sales representative can sell: term, whole life, and universal life. The following matrix gives the number of policies sold by three different representatives during a recent month Whole Universal Life Life 40 Term representaive A 30 representaive B 24 representaive C 34 8 10 32 6 38 Assume representatives A, B, and C earned $4400, $4060, and $4160, respectively, in commissions for these sales. The commission earned depends only on the type of policy sold and is the same for all representatives (10) Use the inverse matrix method to find the commission that is earned for selling a policy of each type. Now use Cramer's rule to find your answer. (10)

Answer 1

solution (i)

Given matrix [x] matrix [bo] : matrix [a] Term whole Universal . life life A [ 30 8 40 ] 24 10 327 [34 6 38 / 44007 . B . Nox 4060 [ 4160 [a] [x] = [b] No. of sells Wil d per unit Commission total caried commission on noun [X] [a]"[b] Грея 4+ . Co- um58709 7 We total earned I commission selis 17) 1 30 8 407744007 OB G 24 10 3240601 12 [ 34 6 38] [ 460 now calculate inueise [a] [a] - [ 30 8 40 1032 L 346 38 inverse of [a] E n - x Adjoint (a). determinant [a] determinant T 30 8 407 24 10 32 3469 38 38 ) zeijo es 1 + **c13"complete ca [[a] = -792]

## usun cofесtvті «Хивті та vic C +52 71 (-) T2 71 L-4 26 1 LG 38 L34 61 # now cofactors of minors mateix Curs E.)T*'728 33 ) = [tex38 - 32 x6] = 188 412= 649 327 = - (21x38- 32x34) = 176 413 = 6,3497 24 10 (2446 – 10 x 34) = -196 C24 = 6,3947 - (8x38 -6X40) = 1-64 Cz2 = 2134 | 30 4072 (30x38 - 34X4D) = -.2.70 193. = 380og i '- (30x6– Duxe) = 92 Cs1 = (1 897 ) = (8x32 - Woxio) = -144 632. = 6,990 291 +(3032 – 40X34) = 0 695 = to 3* Moa (3oxca' = 2428) = 108 L 34 38 Matrix of nie met matrix of minory cofactor = 188. -64 1761-1967 -220 92 o 108 -1894 Adjoint of [a] - transpose of minors cafe ctor mator's

188 Adjoint -64 - 144 -2200 92 108] 1 -196 now invers [a] = I x Adj [a] Tapi I (792) 188 176 7-196 -64 -1447 -220 92 108/ 792 #how per unit comú üssten 17 -188/192 y 12.1 - 176/92 - 1 196/92 67142 14992 1 [ 44007 220/92 o - 9/192 - 108/192 4066 416ol 4400 + GY 792" X 4060 + 144 792 7-176x4400 + 2 x 4060 + o X4400 + 4060) +(- 196 792 4760 407 commission earned by A, B, C is $40, $150 $ 50.

solution (ii)

, Sej (ii) Cramer's rule Given Ta] + [x]. . 30 8 407 / x7 124 10 32 41 | 34 6 381.z! [6] 14400] -4060 (4160 determinant of [a] A = -742) Solve in sejci) the Replace Solution the 1st column of main matrix [a] with vector [b ] { find determinant 401 1,44008 4,060 | H160 38 ello 4066 32 60446010 32 + -1) + (-0) (8) 1 +3 4060 101 40 | 416061 !! 381 478 = 4400 (380 - 192.) - 8(154280 – 133120) + 40(24360 - 41600) = 4400(188) - 8(2-1160) + 40(-17240) 887200 - 1692.80 - 689600 - -31680 and column of matrix [a] with the now replace matrix [b] A2 = 1 30 4400 401 244060 32. 1 34 4160 381 32 4,5 30 / 4060 4060 / ayyoo/ + 40/24 4160 . 38 134 38/ 734 4160

42 = 30 (4060X38 - 32X4160)- 4400(24x38 - 32 X 34 )+40(24x4160-4060X34) A2 = -118800 I now the matrix [b] replace the [a] with 3 Column of matrix 30 @ 4400 6 43 = 30 10 40601 | 24 - 8 4060 + 4400/24 6 4760 34 34 4160 1 34 61 Az = 30 (10x4160 - 4060X6) - 8(2484166-4060x34) +4400(24X6 - 10x34) A3 = -39600) # Solution . - y 4, Z 43 4, x= y = 0 2-43 z = a 2 put values of determinants - -31680 = 40 - 792 now - 1/8800 792 . 150 - - - 39600 - 50 - 792 Solution a = 40 y = 150 = 50 111