Solving a System of Linear Equations — Cramer's Rule and the Matrix Inversion Method

Matrices give two clean, systematic methods for solving simultaneous linear equations — both routinely used in business problems such as finding unit costs or allocating resources from several linear constraints at once.

Cramer's Rule

For two equations in $x,y$:

Form the coefficient determinant $D=\begin{vmatrix}a_1&b_1\\a_2&b_2\end{vmatrix}$, and $D_x, D_y$ by replacing the $x$-column (respectively $y$-column) of $D$ with the constants column $\begin{pmatrix}c_1\\c_2\end{pmatrix}$. Then, provided $D\ne0$,

The same pattern extends to three equations in $x,y,z$: form $D$ from the coefficients of $x,y,z$, and $D_x,D_y,D_z$ by replacing the respective column with the constants, then $x=D_x/D$, $y=D_y/D$, $z=D_z/D$. If $D=0$, Cramer's Rule does not give a unique solution — the system is either inconsistent or has infinitely many solutions, and needs separate treatment.

Matrix Inversion Method

The same system can be written in matrix form $AX=B$, where $A$ is the coefficient matrix, $X$ is the column of unknowns, and $B$ is the column of constants — e.g. for two variables,

Provided $A$ is non-singular, multiply both sides on the left by $A^{-1}$:

$A^{-1}AX = A^{-1}B \implies X = A^{-1}B$ …