. . 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 - - Applications of Matrices and Determinants Then we get AX m m m mn a x a x a x a x a x a x a x a x m m m mn m B Then AX B is a matrix equation involving matrices and it is called the matrix form of the system of linear equations ( ). The matrix A is called the coefficient matrix of the system and the matrix m m m mn m is called the augmented matrix of the system.
We denote the augmented matrix by A B