The transpose A^{T} of a matrix A is the exchange of rows and columns of the matrix.
Example Let
A = [  1  1  2  ], B = [  1  0  1  ], C = [  1  2  ], D = [  0  1  ], X = [  a  b  c  ] 
3  0  1  2  1  1  3  4  1  2  x  y  z 
Then
A^{T} = [  1  3  ], B^{T} = [  1  2  ], C^{T} = 

, D^{T} = 

, X^{T} = [  a  x  ]  
1  0  0  1  b  y  
2  1  1  1  c  z 
In general, if we write a matrix in a compact form A = (a_{ij}), then A^{T} = (a_{ji}).
A = [  a_{11}  a_{12}  ...  a_{1n}  ] ⇒ A^{T} = [  a_{11}  a_{21}  ...  a_{m1}  ] 
a_{21}  a_{22}  ...  a_{2n}  a_{12}  a_{22}  ...  a_{m2}  
:  :  :  :  :  :  
a_{m1}  a_{m2}  ...  a_{mn}  a_{1n}  a_{2n}  ...  a_{mn} 
Moreover, if A is an m by n matrix, then A^{T} is an n by m matrix. The rows of A are the columns of A^{T}, and the columns of A are the rows of A^{T}.
A matrix A is called symmetric if A^{T} = A. It is called skewsymmetric if A^{T} =  A. The matrix D in the example above is symmetric. Clearly, a symmetric (or skewsymmetric) matrix must be a square matrix.
Example The following are the general forms of 2 by 2 and 3 by 3 symmetric matrices.
[  a  b  ] 
b  c 
[  a  b  c  ] 
b  d  e  
c  e  f 
The following are the general forms of 2 by 2 and 3 by 3 skewsymmetric matrices.
[  0  a  ] 
a  0 
[  0  a  b  ] 
a  0  c  
b  c  0 
The transpose has the following properties. The linearity and involutive property are quite obvious. The product property is not so obvious.
The properties can also be explained by the interpretation of the transpose in terms of the linear transformations. In the equivalence between matrices and linear transformations, we would like to find the answer to 4^{?} in the following table.
linear transformation  matrix 
composition  multiplication 
addition  addition 
scalar multiplication  scalar multiplication 
inverse  inverse 
4^{?}  transpose 
The answer is the dual of linear transformations. The linearity and the product properties are quite easy to see from the transformation viewpoint. The interpretation of the involutive property is somewhat complicated and can be found here.