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 | ... | a2n | x2 | b2 | |||||||||
| ······························ | ··· | ··· | ||||||||||||
| am1 | am2 | ... | amn | xn | bm | |||||||||
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 | x2 | 5 | ||||||||
| 6 | 2 | 4 | 0 | x3 | 2 | ||||||||
| 0 | 2 | -1 | 1 | x4 | 0 |
MatrixMatrix Definition. Main informationSystem of linear equations - matrix formTypes of matricesMatrix scalar multiplicationAddition and subtraction of matricesMatrix multiplicationTranspose matrixElementary matrix operationsDeterminant of a matrixMinors and cofactors of a matrixInverse matrixLinearly dependent and independent rowsRank of a matrix
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