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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

0

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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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