The transpose of a matrix.
.
These are examples of matrix transpose. In general, for a matrix 
%7Bm%20%5Ctimes%20n%7D#card=math&code=A%3D%28a%7Bij%7D%29%7Bm%20%5Ctimes%20n%7D), the transpose of 
, we denote by 
, is an 
matrix. And the 
#card=math&code=%28i%2C%20j%29)-entry of 
is equal to .
Example: The (3, 1) entry of 
is the 
#card=math&code=%281%2C%203%29) entry of 
.
A few properties for transpose.
(1) 
%5E%5Ctop%20%3D%20A#card=math&code=%28A%5E%5Ctop%29%5E%5Ctop%20%3D%20A).
(2) 
%5E%7B%5Ctop%7D%20%3D%20A%5E%5Ctop%20%2BB%5E%5Ctop#card=math&code=%28A%2BB%29%5E%7B%5Ctop%7D%20%3D%20A%5E%5Ctop%20%2BB%5E%5Ctop).
(3) 
%5E%5Ctop%20%3D%20%5Calpha%20A%5E%5Ctop#card=math&code=%28%5Calpha%20A%29%5E%5Ctop%20%3D%20%5Calpha%20A%5E%5Ctop).
(4) Suppose 
is well-defined. Then 
%5E%5Ctop%20%3D%20B%5E%5Ctop%20A%5E%5Ctop#card=math&code=%28AB%29%5E%5Ctop%20%3D%20B%5E%5Ctop%20A%5E%5Ctop).
(check this property yourself)
(5) For a square matrix , 
%20%3D%20%5Cdet%20(A%5E%5Ctop)#card=math&code=%5Cdet%20%28A%29%20%3D%20%5Cdet%20%28A%5E%5Ctop%29).
Example exercise on transpose of matrices.
We assume that 
. Then 
%20(x%2C%20y%2C%20z)%5E%5Ctop%20%3D%20x-2z#card=math&code=0%3D%281%20%2C%200%20%2C%20-2%29%20%28x%2C%20y%2C%20z%29%5E%5Ctop%20%3D%20x-2z). so 
.
so that 
and 
.
. 
and hence 
.
Comments on hw Prob. 349, 350 (Exercise 5.6)
Prob. 350.
‘%20%3D%20%5Cfrac%7B1%7D%7B%5Ccos%20x%5E2%7D%20(%5Ccos%20x%5E2)’%3D(-%5Csin%20x%5E2%2F%5Ccos%20x)%20(2x)%20%3D%20-%5Cfrac%7B2x%20%5Csin%20x%5E2%7D%7B%5Ccos%20x%5E2%7D#card=math&code=%5CLARGE%20%5Cln%20%28%20%5Ccos%20x%5E2%29%27%20%3D%20%5Cfrac%7B1%7D%7B%5Ccos%20x%5E2%7D%20%28%5Ccos%20x%5E2%29%27%3D%28-%5Csin%20x%5E2%2F%5Ccos%20x%29%20%282x%29%20%3D%20-%5Cfrac%7B2x%20%5Csin%20x%5E2%7D%7B%5Ccos%20x%5E2%7D)
Prob. 349
%2B%5Ccos(3x))’%20%3D%20%5Ccos(3x)3%20-%20%5Csin(3x)3%3D-3(%5Csin(3x)-%5Ccos(3x))#card=math&code=%28%5Csin%20%283x%29%2B%5Ccos%283x%29%29%27%20%3D%20%5Ccos%283x%29%2A3%20-%20%5Csin%283x%29%2A3%3D-3%28%5Csin%283x%29-%5Ccos%283x%29%29)
The answer is 
%2B%5Ccos(3x)%7C%2BC#card=math&code=%5Cfrac%7B1%7D%7B-3%7D%20%5Cln%20%7C%5Csin%20%283x%29%2B%5Ccos%283x%29%7C%2BC)
