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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