# Coefficient matrix

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Matrix whose entries are the coefficients of a linear equation

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In [linear algebra](/source/Linear_algebra), a **coefficient matrix** is a [matrix](/source/Matrix_(mathematics)) consisting of the [coefficients](/source/Coefficient) of the variables in a set of [linear equations](/source/Linear_equations). The matrix is used in solving [systems of linear equations](/source/Systems_of_linear_equations).

## Coefficient matrix

In general, a system with m [linear equations](/source/Linear_equations) and n unknowns can be written as

- a 11 x 1 + a 12 x 2 + ⋯ + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + ⋯ + a 2 n x n = b 2 ⋮ a m 1 x 1 + a m 2 x 2 + ⋯ + a m n x n = b m {\displaystyle {\begin{aligned}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&=b_{1}\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&=b_{2}\\&\;\;\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&=b_{m}\end{aligned}}}

where x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} are the unknowns and the numbers a 11 , a 12 , … , a m n {\displaystyle a_{11},a_{12},\ldots ,a_{mn}} are the coefficients of the system. The coefficient matrix is the *m* × *n* matrix with the coefficient aij as the (*i, j*)th entry:[1]

- [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] {\displaystyle {\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}}}

Then the above set of equations can be expressed more succinctly as

- A x = b {\displaystyle A\mathbf {x} =\mathbf {b} }

where A is the coefficient matrix and **b** is the column vector of constant terms.

## Relation of its properties to properties of the equation system

By the [Rouché–Capelli theorem](/source/Rouch%C3%A9%E2%80%93Capelli_theorem), the system of equations is [inconsistent](/source/Inconsistent_equations), meaning it has no solutions, if the [rank](/source/Rank_(linear_algebra)) of the [augmented matrix](/source/Augmented_matrix) (the coefficient matrix augmented with an additional column consisting of the vector **b**) is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank r equals the number n of variables. Otherwise the general solution has n – r free parameters; hence in such a case there are an infinitude of solutions, which can be found by imposing arbitrary values on n – r of the variables and solving the resulting system for its unique solution; different choices of which variables to fix, and different fixed values of them, give different system solutions.

## Dynamic equations

A first-order [matrix difference equation](/source/Matrix_difference_equation) with constant term can be written as

- y t + 1 = A y t + c , {\displaystyle \mathbf {y} _{t+1}=A\mathbf {y} _{t}+\mathbf {c} ,}

where A is *n* × *n* and **y** and **c** are *n* × 1. This system converges to its steady-state level of y [if and only if](/source/If_and_only_if) the [absolute values](/source/Absolute_value) of all n [eigenvalues](/source/Eigenvalue) of A are less than 1.

A first-order [matrix differential equation](/source/Matrix_differential_equation) with constant term can be written as

- d y d t = A y ( t ) + c . {\displaystyle {\frac {d\mathbf {y} }{dt}}=A\mathbf {y} (t)+\mathbf {c} .}

This system is stable if and only if all n eigenvalues of A have negative [real parts](/source/Complex_number).

## References

1. **[^](#cite_ref-Liebler_1-0)** Liebler, Robert A. (December 2002). [*Basic Matrix Algebra with Algorithms and Applications*](https://books.google.com/books?id=dD1OKMD-rMoC&q=coefficient+matrix+linear+systems). [CRC Press](/source/CRC_Press). pp. 7–8. [ISBN](/source/ISBN_(identifier)) [9781584883333](https://en.wikipedia.org/wiki/Special:BookSources/9781584883333). Retrieved 13 May 2016.

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