Processing math: 100%

A.Kaw矩阵代数初步学习笔记 4. Unary Matrix Operations

“矩阵代数初步”(Introduction to MATRIX ALGEBRA)课程由Prof. A.K.Kaw(University of South Florida)设计并讲授。
PDF格式学习笔记下载(Academia.edu)
第4章课程讲义下载(PDF)

Summary

  • Transpose
    Let [A] be a m×n matrix. Then [B] is the transpose of [A] if bji=aij for all i and j. That is, the i-th row and the j-th column element of [A] is the j-th row and i-th column element of [B]. Note that [B] is a n×m matrix and is denoted by [B]=[A]T. For example, [A]=[123456][A]T=[142536]
  • Symmetric matrix
    A square matrix [A] with real elements where aij=aji for i=1,,n and j=1,,n is called a symmetric matrix. That is, [A] is a symmetric matrix if [A]=[A]T. For example, [A]=[123245357]
  • Skew-symmetric matrix
    A n×n matrix is skew-symmetric if aij=aji for i=1,,n and j=1,,n. That is, [A] is a skew-symmetric matrix if [A]=[A]T. Note that the diagonal elements must be zero in a skew-symmetric matrix. For example, [A]=[023205350]
  • Trace of matrix
    The trace of a n×n matrix [A] is the sum of the diagonal entries of [A], that is, tr[A]=ni=1aii For example, [A]=[123245357]tr[A]=1+4+7=12
  • Determinant
    Let [A] be a n×n matrix.
    • The minor of entry aij is denoted by Mij and is defined as the determinant of the (n1)×(n1) sub-matrix of [A], where the sub-matrix is obtained by deleting the i-th row and j-th column of the matrix [A]. The determinant is then given by det(A)=nj=1(1)i+jaijMij, for any i=1,2,,n or det(A)=ni=1(1)i+jaijMij, for any j=1,2,,n For example, [A]=[123245357] det(A)=(1)1+11|4557|+(1)1+22|2537|+(1)1+33|2435| =(4×75×5)2×(2×73×5)+3×(2×53×4)=1 Note that for a 2×2 matrix [A]=[abcd], det(A)=adbc.
    • The number (1)i+jMij is called the cofactor of aij and is denoted by Cij. The formula for the determinant can then be written as det(A)=nj=1aijCij, for any i=1,2,,n or det(A)=ni=1aijCij, for any j=1,2,,n
    • If [A] and [B] are square matrices of same size, then det(AB)=det(A)det(B)
    • det(A)=0 if
      • A row or a column is zero, or
      • A row (column) is proportional to another row (column).
    • If a row (column) is multiplied by k to result in matrix [B], then det(B)=kdet(A)
    • If [B]=k[A], then det(B)=kndet(A)
    • If [A] is a n×n upper or lower triangular matrix, then det(A)=ni=1aii
    • If [B] is row-equivalent to [A], then {RiRj:det(B)=det(A);tRi:det(B)=tdet(A);RiRi+tRj:det(B)=det(A).

Selected Problems

1. Let [A]=[2536792] Find [A]T.

Solution:

[A]T=[2573962]

2. If [A] and [B] are two n×n symmetric matrices, show that [A]+[B] is also symmetric.

Solution:

Let [C]=[A]+[B], so we have cij=aij+bij=aji+bji=cji that is, [C]=[C]T.

3. What is the trace of [A]=[72345555667952310]

Solution:

tr[A]=75+7+10=19

4. Find the determinant of [A]=[107032.0996515]

Solution:

det(A)=(1)1+1×10×|2.099615|+(1)1+2×(7)×|3655| =10×(2.099×5+1×6)+7×(1530)=150.05

5. What is the value of a n×n matrix det(3[A])?

Solution:

det(3[A])=3ndet(A)

6. For a 5×5 matrix [A], the first row is interchanged with the fifth row, what is the determinant of the resulting matrix [B]?

Solution:

The sign would be changed if interchaged two row (column). Thus det(B)=det(A)

7. What is the determinant of [A]=[0100001000011000]

Solution:

[A]=[0100001000011000]R1R4[1000001000010100] R2R3[1000000100100100] R2R4[1000010000100001]=[B] Thus det(A)=(1)3det(B)=1.

8. Find the determinant of [A]=[000235692]

Solution:

det(A)=0 since the first row is zero.

9. Find the determinant of [A]=[0023023567236.67.72.23.3]

Solution:

Since R4=1.1R3, so det(A)=0.

10. Find the determinant of [A]=[5000030025601239]

Solution:

This is a lower triangular matrix and hence det(A)=5×3×6×9=810

11. Given the matrix [A]=[125255151264811157891318421] and det(A)=32400. Find the determinant of [A1]=[12525515126481114181918421]; [A2]=[125251551264181157891138412]; [A3]=[125255111578913151264818421]; [A4]=[125255111578913184215126481]; [A5]=[1252551512648111578913116842].

Solution:

[A]=[125255151264811157891318421]R32R4[12525515126481114181918421]=[A1] Thus det(A1)=det(A)=32400. [A]=[125255151264811157891318421]C3C4[125251551264181157891138412]=[A2] Thus det(A2)=det(A)=32400. [A]=[125255151264811157891318421]R2R3[125255111578913151264818421]=[A3] Thus det(A3)=det(A)=32400. [A]=[125255151264811157891318421]{R2R3R3R4[125255111578913184215126481]=[A4] Thus det(A4)=(1)2det(A)=32400. [A]=[125255151264811157891318421]2R4[1252551512648111578913116842]=[A5] Thus det(A5)=2det(A)=64800.

12. Find the determinant of [A]=[25516481144125]

Solution:

det(A)=(1)1+3a13M13+(1)2+3a23M23+(1)3+3a33M33 =|64814412||25514412|+5×|255648|=564

13. Show that if [A][B]=[I], where [A], [B] and [I] are matrices of n×n size and [I] is an identity matrix, then det(A)0 and det(B)0.

Solution: det(A)det(B)=det(AB)=det(I)=1 det(A)0, det(B)0.

14. If the determinant of a 4×4 matrix [A] is given as 20, then what is the determinant of 5[A]?

Solution:

det(k[A])=kndet(A) det(5[A])=54det(A)=625×20=12500

15. If the matrix product [A][B][B] is defined, what is ([A][B][C])T?

Solution:

([A][B])T=[B]T[A]T ([A][B][C])T=[C]T([A][B])T=[C]T[B]T[A]T

16. The determinant of the matrix [A]=[255103809a] is 50. What is the value of a?

Solution:

det(A)=25×|389a|=25×(3a72)=50 a=743

17. [A] is a 5×5 matrix and a matrix [B] is obtained by the row operations of replacing Row1 with Row3, and then Row3 is replaced by a linear combination of 2×Row3+4×Row2. If det(A)=17, then what is the value of det(B)?

Solution:

The process is [A]R1R32R3R3+4R2[B] Thus det(B)=(1)×2det(A)=34

posted on   赵胤  阅读(339)  评论(0编辑  收藏  举报

导航

统计

点击右上角即可分享
微信分享提示