System of linear equations - matrix form
Any system of linear equations can be written as the matrix equation.
So a system of linear equations
a_{11}x_{1} + a_{12}x_{2} + ... + a_{1n}x_{n} = b_{1} | |
a_{21}x_{1} + a_{22}x_{2} + ... + a_{2n}x_{n} = b_{2} | |
································ | |
a_{m1}x_{1} + a_{m2}x_{2} + ... + a_{mn}x_{n} = b_{m} |
consisting of m linear equations with n unknowns can be written as a matrix equation:
Ax = b
where
A = | a_{11} | a_{12} | ... | a_{1n} | ; x = | x_{1} | ; b = | b_{1} | ||||||
a_{21} | a_{22} | ... | a_{2n} | x_{2} | b_{2} | |||||||||
······························ | ··· | ··· | ||||||||||||
a_{m1} | a_{m2} | ... | a_{mn} | x_{n} | b_{m} |
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 x_{j}, it means that it multiplier is zero, ie a_{ij} = 0.
Example of matrix form of system of linear equations
Example 1.
Write system of linear equations in matrix form:
4x_{1} + x_{2} - x_{3} - x_{4} = 3 | |
-x_{1} + 3x_{3} - 2x_{4} = 5 | |
6x_{1} + 2x_{2} + 4x_{3} = 2 | |
2x_{2} - x_{3} + x_{4} = 0 |
Solution: System of linear equations in matrix form:
4 | 1 | -1 | -1 | · | x_{1} | = | 3 | ||||||
-1 | 0 | 3 | -2 | x_{2} | 5 | ||||||||
6 | 2 | 4 | 0 | x_{3} | 2 | ||||||||
0 | 2 | -1 | 1 | x_{4} | 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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