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System of linear equations - matrix form

Any system of linear equations can be written as the matrix equation.

So a system of linear equations

{ a11x1 + a12x2 + ... + a1nxn = b1
a21x1 + a22x2 + ... + a2nxn = b2
································
am1x1 + am2x2 + ... + amnxn = bm

consisting of m linear equations with n unknowns can be written as a matrix equation:

Ax = b

where

A =  (  a11  a12  ...  a1n ) ;     x(x1) ;     b(b1)
 a21  a22  ...  a2nx2b2
····································
 am1  am2  ...  amnxnbm

Matrix A is the matrix of coefficient of a system of linear equations, the column vector x is vector of unknowns variables, and the column vector b is vector of a system of linear equations values.

N.B. If the i-th row of the system of linear equations is not the variable xj, it means that it multiplier is zero, ie aij = 0.

Example of matrix form of system of linear equations

Example 1.
Write system of linear equations in matrix form:
{ 4x1 + x2 - x3 - x4 = 3
-x1 + 3x3 - 2x4 = 5
6x1 + 2x2 + 4x3 = 2
2x2 - x3 + x4 = 0

Solution: System of linear equations in matrix form:

(  4  1  -1  -1  )  · (x1)  = (3)
 -1  0  3  -2 x25
 6  2  4  0 x32
 0  2  -1  1 x40

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