Page 29 - mathematics

Transpose of a matrix: Example

A  1 0 2 , B  2 1 1 , 5
3 2 1 2 0 3 C  0

3

Solution

a11 a21  am1  1 3 T 1 1
a12  AT  0 2 , 1
AT   a22  am2  1 A  BT 1 1 1  1 2  ,
2  1 2 2 
a1n  

a2n  amn  2

CT  5 0 3.

Properties of transpose of a matrix: Special Matrices
(AT )T  A
1) Zero matrix 0: 0 0 0 0 0
(A  B)T  AT  BT 0  0 0  0 0 0 0
(AB)T  BT AT 0 0 
0 0
( A)T   AT
A  0  A, A  (A)  0
(A  B  C)T  AT  BT  CT
2) Triangular matrix:
(ABC)T  CT BT AT
1 2 3 4 -2 0 0 0

U  0 5 6 7 L   1 6 0 0
0 0 8 9  8 9 3 0


0 0 0 1  4 2 1 5


upper triangular matrix lower triangular matrix

3) Diagonal matrix: 6) Symmetric matrix:
7 0 0
D  0 0
1 AT  A; 1 2 7
2 A  2 5 6
0 0 1
7 6 4
4) Scalar matrix:

5 0 7) Skew symmetric:
0 5
 0 1 1
5) Identity matrix: AT  A;
1 0 0 A  1 0 2 

I  0 1 0  1 2 0 
0 0 1

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