System of Linear Equations in Matrix Form

System of Linear Equations in Matrix Form

A system of m linear equations in n unknowns is of the following form:

a₁₁x₁ + a₁₂x₂ + a₁₃x₃ + ……… + a_1nx_n + = b₁

a₂₁x₁ + a₂₂x₂ + a₂₃x₃ + ……… + a_2nx_n + = b₂

a₃₁x₁ + a₃₂x₂ + a₃x₃ + ……… + a_3nx_n + = b₃

…..    â€¦.   â€¦..   â€¦..   â€¦..    ...

A_m1x₁ + a_m2x₂ + a_m3x₃ + ……… + a_mnx_n + = b_m

where the coefficients  a_ij , i = 1, 2, …. , m; j = 1, 2,….., n  and b_k , k = 1, 2,….., m   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 α₁ , α₂ , ….. , α_n for x₁, x₂ , …. , x_n respectively which satisfy every  equation of (1), then the ordered  n − tuple (α₁ ,  Î±₂ , …. , α_n ) is called a solution of (1). The above system (1) can be put in a matrix form as follows:

Let A =  be   the m x n matrix formed by the coefficients of x₁, x₂ , x₃,…. , x_n . The first row of A is formed by the coefficients of x₁, x₂ , x₃,…. , 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 x₁ in the m equations in the same order. The other columns are formed in a similar way.

Let  X =  be the n x1 order column matrix formed by the unknowns x₁, x₂ , x₃,…. , x_n

Let   B  =  be the m x 1 order column matrix formed by the right-hand side constants b₁, b₂ , b₃ , …. , b_m .

Then we get

Then AX = B is a matrix equation involving matrices and it is called the matrix form of the system of linear equations (1). The matrix A is called the coefficient matrix of the system and the matrix

is called the augmented matrix of the system. We denote the augmented matrix by [ A | B ].

As an example, the matrix form of the system of linear equations

2x + 3y - 5z + 7 = 0, 7 y + 2z - 3x = 17, 6x - 3y - 8z + 24 = 0 is