1.4.2 System of Linear Equations in Matrix Form

A system of m linear equations in n unknowns is of the following form: a x a x a x a x a x a x a x a x          a x m m m m m … ( ) where the coefficients a i m j ij , , , ; , ,   and b k m k , , , =  are constants. If all the b k 's are zeros, then the above system is called a homogeneous system of linear equations. On the other hand, if at least one of the b k 's is non-zero, then the above system is called a non-homogeneous system of linear equations. If there exist values α  n for x x n  respectively which satisfy every equation of ( ), then the ordered n − tuple α  ) is called a solution of ( ).

The above system ( ) can be put in a matrix form as follows: Let A        m m m mn be the m n matrix formed by the coefficients of x n , ,  .The first row of A is formed by the coefficients of x x n , ,  in the same order in which they occur in the first equation. Likewise, the other rows of A are formed. The first column is formed by the coefficients of x in the m equations in the same order. The other columns are formed in a similar way.

Let X x n  be the n × order column matrix formed by the unknowns x x n , ,  Let B b m  be the m × order column matrix formed by the right-hand side constants b m 