Direct Method of Interpolation: General Engineering



Problem Set

Chapter 04.04

Unary Matrix Operations

1. Let

[pic].

Find [pic]

2. If [pic] and [pic] are two [pic] symmetric matrices, show that [pic] is also symmetric. Hint: Let [pic]

3. Give an example of a [pic] symmetric matrix.

4. Give an example of a [pic] skew-symmetric matrix.

5. What is the trace of

A) [pic]

B) For

[pic]

Find the determinant of [pic] using the cofactor method.

6. [pic] of a [pic] matrix is

A) [pic]

B) 3[pic]

C) [pic]

D) [pic]

7. For a [pic] matrix [pic], the first row is interchanged with the fifth row, the determinant of the resulting matrix [pic]is

A) [pic]

B) [pic]

C) [pic]

D) [pic]

8. [pic] is

A) 0

B) 1

C) –1

D) (

9. Without using the cofactor method of finding determinants, find the determinant of

[pic]

10. Without using the cofactor method of finding determinants, find the determinant of

[pic]

11. Without using the cofactor method of finding determinants, find the determinant of

[pic]

12. Given the matrix

[pic]

and

[pic]

find the determinant of

A) [pic]

B) [pic]

C) [pic]

D) [pic]

E) [pic]

13. What is the transpose of

[pic]

14. What values of the missing numbers will make this a skew-symmetric matrix?

[pic]

15. What values of the missing number will make this a symmetric matrix?

[pic]

16. Find the determinant of

[pic]

17. What is the determinant of an upper triangular matrix [pic] that is of order [pic]?

18. Given the determinant of

[pic]

is[pic], find [pic].

19. Why is the determinant of the following matrix zero?

[pic]

20. Why is the determinant of the following matrix zero?

[pic]

21. Show that if [pic], where [pic], [pic] and [pic] are matrices of [pic] size and [pic] is an identity matrix, then [pic] and [pic].

Answers to Selected Problems

1. [pic]

2. [pic] for all i, j.

and

[pic] for all i, j.

[pic] as [pic] and [pic] are symmetric

Hence [pic]

3.

4.

5. a)[pic]

b) [pic]

6. C

7. A

8. C

9. 0: Can you answer why?

10. 0: Can you answer why?

11. [pic]: Can you answer why?

12. –32400 b) 32400 c) 32400 d) –32400 e) –64800

13. [pic]

14. [pic]

15. [pic]

16. The determinant of [pic] is

[pic]

[pic]

[pic]

17. The determinant of an upper triangular matrix is the product of its diagonal elements,[pic]

18. [pic]

[pic]

[pic]

19. The first row of the matrix is zero, hence, the determinant of the matrix is zero.

20. Row 4 of the matrix is 1.1 times Row 3. Hence, its determinant is zero.

21. We know that det(AB) = det(A)det(B).

[A][B] = [I]

det(AB) = det(I)

det(I) = [pic]

det(A)det(B) = 1

Therefore,

[pic] and

[pic].

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