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Matrix transpose

Definition.
Transposing the matrix is an operation on the matrix in which its rows and columns are swapped:

aTij = aji


Properties of transpose matrix

  • If matrix A has a size of n×m, then the transposed matrix AT has a size of m×n;
  • (AT)T = A;
  • (k · A)T = k · AT;
  • (A + B)T = AT + BT;
  • (A · B)T = BT · AT.

Examples of matrix transpose

Example 1.
Find the transposed matrix AT for matrix
A =  (  4  2  ) .
 9  0 

Solution:

AT (  4  9  )
 2  0 
Example 2
Find the transposed matrix AT for matrix
A =  (  2  1  ) .
 -3  0 
 4  -1 

Solution:

AT (  2  -3  4  )
 1  0  -1 
Example 3
Find the transposed matrix AT for matrix
A =  (  2  -3  4  ) .
 1  0  -1 

Solution:

AT (  2  1  )
 -3  0 
 4  -1 

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