Norwich University Mathematical Economics Questions

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1. Evaluate the following determinants:
| 8 7 3
4 0 2
(a) 4 0 1
(c) 6 0 3
6 0 3
8 2 3
la b
(e) b c
C
b
O co-
? ?? *
ç
1
1 2 3
1 4
5 0
(b) 4 7 5
(d) 8 11 -2 () 3 y 2
3 6 9
4 7
9
-18
2. Determine the signs to be attached to the relevant minors in order to get the following
cofactors of a determinant: 1613, C231, 1C331, C411, and 1C341.
a b
3. Given de , find the minors and cofactors of the elements a, b, and f.
| g h i
4. Evaluate the following determinants:
1 2 0 9
2 7 0 1
2 3 4 6
5 6 4 8
(a)
(b)
1 6 0 -1
0 090
0-50 8
1 -3 1 4
5. In the first determinant of Prob. 4, find the value of the cofactor of the element 9.
6. Find the minors and cofactors of the third row, given
9 11 4
? 3 27
6 10 4
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EXERCISE 5.5
1. Use Cramer’s rule to solve the following equation systems:
(0) 3×1 – 2×2 = 6
(C) 8×1 – 7×2 = 9
2×1 + x2 = 11
X1 + X2 = 3
(b) – x1 + 3×2 = -3 (d) 5×1 to 9×2 = 14
4×1 – x2 = 12
7×1 – 3×2
4
2. For each of the equation systems in Prob. 1, find the inverse of the coefficient matrix,
and get the solution by the formula x* = A-ld.
3. Use Cramer’s rule to solve the following equation systems:
(a) 8×1
16 (c) 4x + 3y – 2z=1
2×2 + 5×3
x + 2y
2×1
+ 3×3
7
3x + Z=4
(6) – x1 + 3×2 + 2×3 = 24 (d) -x + y + z=a
X to X3 : 6
x-y+z= b
5×2 – x3 = 8 x+y-7=C
4. Show that Cramer’s rule can be derived alternatively by the following procedure. Mul-
tiply both sides of the first equation in the system Ax = d by the cofactor Chil, and
then multiply both sides of the second equation by the cofactor Cz;l, etc. Add all the
newly obtained equations. Then assign the values 1, 2, …, n to the index }, SUCCES-
sively, to get the solution values **, * , , ,x as shown in (5.17).
– X2

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EXERCISE 5.4
]
[-2]
[3 -1]
[]
9
3
1. Suppose that we expand a fourth-order determinant by its third column and the cofac-
tors of the second-column elements. How would you write the resulting sum of prod-
ucts in notation? What will be the sum of products in notation if we expand it by
the second row and the cofactors of the fourth-low elements?
2. Find the inverse of each of the following matrices;
[5 27
-1 01
71
7 6
(a) A =
(b) 8 =
01
(C) C=
(d) D
03
3. (a) Drawing on your answers to Prob. 2, formulate a two-step rule for finding the ad-
joint of a given 2 x 2 matrix A: In the first step, indicate what should be done to the
two diagonal elements of A in order to get the diagonal elements of adj A; in the
second step, indicate what should be done to the two off-diagonal elements of A.
(Warning: This rule applies only to 2 x 2 matrices.)
(b) Add a third step which, in conjunction with the previous two steps, yields the 2 x 2
inverse matrix A-1.
4. Find the inverse of each of the following matrices:
4 -21
100
(a) E = 7 3 0
(0) G=
0 0
0
0 1 0
1
2
1
(6) F
-1 2
03
02
1
4
(d) H =
1 0 0
0 1 0
0 0 1
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Chapter 5 l.inear Models and Marrix Algebra (Conrinued) 103
5. Find the inverse of
A-
4 1
-2 3
3 -1
1
4
6. Solve the system Ax = d by matrix inversion, where
(a) 4x + 3y – 28
(b) 4×1 + x2 – 5×3
2x + 5y = 42
-2x + 3×2 + x3
3X; – x2 + 4×3
7. Is it possible for a matrix to be its own inverse?
12

5
.
5.5 Cramer’s Rule
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EXERCISE 5.5
1. Use Cramer’s rule to solve the following equation systems:
(0) 3×1 – 2×2 = 6
(C) 8×1 – 7×2 = 9
2×1 + x2 = 11
X1 + X2 = 3
(b) – x1 + 3×2 = -3 (d) 5×1 to 9×2 = 14
4×1 – x2 = 12
7×1 – 3×2
4
2. For each of the equation systems in Prob. 1, find the inverse of the coefficient matrix,
and get the solution by the formula x* = A-ld.
3. Use Cramer’s rule to solve the following equation systems:
(a) 8×1
16 (c) 4x + 3y – 2z=1
2×2 + 5×3
x + 2y
2×1
+ 3×3
7
3x + Z=4
(6) – x1 + 3×2 + 2×3 = 24 (d) -x + y + z=a
X to X3 : 6
x-y+z= b
5×2 – x3 = 8 x+y-7=C
4. Show that Cramer’s rule can be derived alternatively by the following procedure. Mul-
tiply both sides of the first equation in the system Ax = d by the cofactor Chil, and
then multiply both sides of the second equation by the cofactor Cz;l, etc. Add all the
newly obtained equations. Then assign the values 1, 2, …, n to the index }, SUCCES-
sively, to get the solution values **, * , , ,x as shown in (5.17).
– X2

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