Under this section, we will be learning about important terms which are frequently used in matrices.
We will discuss the following:
•    Transpose of a Matrix
•    Minors
•    Co-factors

Let’s study each one in detail.

Transpose of a Matrix

Let A be a m x n matrix, then its transpose is obtained by interchanging rows into columns.  It is denoted by A^T or A’
If A is of order m x n, then the order of A’ is n x m
For example:        Order of A=2x3
                   

      Order of A’=3x2
 

Properties of transpose of the Matrix

For any matrices A and B of suitable orders, we have
1) (A’)’= A
2) (kA)’ = kA’
3) (A + B)’ = A’ + B’
4) (AB)’ = B’A’

Let’s try the following examples:
1) If A= [1   4    5] then show that (A’)’ = A

             
                      
                      
                       
From (i) and (ii), we get, (A + B)’ = A’ + B’
                                     

Minor of an element

Minor of an element a_ij of a determinant is the determinant obtained by deleting its i^th row and j^th column in which a_ij lies.  It is denoted by M_ij. 
Minor of an element of a determinant of order n (n ≥ 2) is a determinant of order n - 1
Example: Find the minor of the element 3 in the determinant

Solution: The element 3 lies in first row and third column, so it is denoted by M₁₃ and is given by
                                                 [Deleting 1st row and 3rd column]
                                    = 0 – (-15)
                                    = 15                  

Co-factor of an element

Co-factor of an element aij denoted by A_ij is defined by A_ij = (-1)i+j M_ij ,  where M_ij is the minor of a_ij.
Example: Find the co-factor of element -5 in the determinant
                                                                                              
                                                                                               
Solution: -5 belongs to 3rd row and 1st column, so it is denoted by

          
      = + (0 - 15)
      = -15
                                     

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Reference Links:

• http://en.wikipedia.org/wiki/Transpose
• http://www.mathreference.com/la-mpoly,order.html
• http://en.wikipedia.org/wiki/Minor_(linear_algebra)
• http://en.wikipedia.org/wiki/Determinant
• http://en.wikipedia.org/wiki/Cofactor_(linear_algebra)