Solving a System of Linear Equations by the Matrix Inversion Method

A system of linear equations can be written compactly in matrix form and solved directly using the matrix inverse — a method that generalises cleanly regardless of how many equations are involved.

Writing a system in matrix form

A system such as

can be written as $AX=B$, where

$A$ is called the coefficient matrix, $X$ the variable matrix, and $B$ the constant matrix. The same idea extends directly to three equations in three unknowns, with $A$ a $3\times3$ coefficient matrix.

Solving for X

If $A$ is non-singular ($|A|\neq0$), multiplying both sides of $AX=B$ on the left by $A^{-1}$ gives

So the unique solution is simply $X=A^{-1}B$ — once $A^{-1}$ is known, solving the whole system is a single matrix multiplication.

What if A is singular?

If $|A|=0$, the matrix-inversion method cannot be used at all ($A^{-1}$ does not exist) — the system either has no solution or infinitely many, and must be examined by other means. This is a genuine limitation of the method, not a computational inconvenience to work around. …