Coefficient matrix

Definition

Let $F$ be a field. Take a system of linear equations over $F$ with $m$ equations in $n$ unknowns, where equation $i$ is $a_{i1} x_1 + \cdots + a_{in} x_n = b_i$. The coefficient matrix of the system is the $m \times n$ matrix $A = (a_{ij})$. Its entry in row $i$ and column $j$ is the coefficient of $x_j$ in equation $i$.

Remarks

The system $2x + y = 5$, $x - 3y = 0$ has the coefficient matrix $\begin{pmatrix} 2 & 1 \\ 1 & -3 \end{pmatrix}$. With $x = (x_1, \dots, x_n)^T$ and $b = (b_1, \dots, b_m)^T$, the system is $Ax = b$. If $A$ is square and invertible, the system has exactly one solution, $x = A^{-1} b$.

Explanations

L1 human

The coefficient matrix keeps only the numbers in front of the unknowns, in a rectangular table. Each row is one equation, and each column is one unknown. This table lets us work with the whole system as one object.