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]=[023−205−3−50] - Trace of matrix
The trace of a n×n matrix [A] is the sum of the diagonal entries of [A], that is, tr[A]=n∑i=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 (n−1)×(n−1) 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)=n∑j=1(−1)i+jaijMij, for any i=1,2,⋯,n or det(A)=n∑i=1(−1)i+jaijMij, for any j=1,2,⋯,n For example, [A]=[123245357] ⇒det(A)=(−1)1+1⋅1⋅|4557|+(−1)1+2⋅2⋅|2537|+(−1)1+3⋅3⋅|2435| =(4×7−5×5)−2×(2×7−3×5)+3×(2×5−3×4)=−1 Note that for a 2×2 matrix [A]=[abcd], det(A)=ad−bc.
- 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)=n∑j=1aijCij, for any i=1,2,⋯,n or det(A)=n∑i=1aijCij, for any j=1,2,⋯,n
- If [A] and [B] are square matrices of same size, then det(A⋅B)=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)=k⋅det(A)
- If [B]=k⋅[A], then det(B)=kndet(A)
- If [A] is a n×n upper or lower triangular matrix, then det(A)=n∏i=1aii
- If [B] is row-equivalent to [A], then {Ri↔Rj:det(B)=−det(A);tRi:det(B)=tdet(A);Ri→Ri+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]=[7234−5−5−5−56679−52310]
Solution:
tr[A]=7−5+7+10=19
4. Find the determinant of [A]=[10−70−32.09965−15]
Solution:
det(A)=(−1)1+1×10×|2.0996−15|+(−1)1+2×(−7)×|−3655| =10×(2.099×5+1×6)+7×(−15−30)=−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]⇒R1↔R4[1000001000010100] ⇒R2↔R3[1000000100100100] ⇒R2↔R4[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]=[125255151264811141819−18421]; [A2]=[125251551264181157891138412]; [A3]=[125255111578913151264818421]; [A4]=[125255111578913184215126481]; [A5]=[1252551512648111578913116842].
Solution:
[A]=[125255151264811157891318421]⇒R3−2R4[125255151264811141819−18421]=[A1] Thus det(A1)=det(A)=−32400. [A]=[125255151264811157891318421]⇒C3↔C4[125251551264181157891138412]=[A2] Thus det(A2)=−det(A)=32400. [A]=[125255151264811157891318421]⇒R2↔R3[125255111578913151264818421]=[A3] Thus det(A3)=−det(A)=32400. [A]=[125255151264811157891318421]⇒{R2↔R3R′3↔R4[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×(3a−72)=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]⇒R1↔R3⇒2R3⇒R3+4R2⇒[B] Thus det(B)=(−1)×2⋅det(A)=−34
作者:赵胤
出处:http://www.cnblogs.com/zhaoyin/
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