Solving a System with the Matrix Inverse

Any system of linear equations can be packed into the single matrix equation . You cannot divide by a matrix — but you can multiply by its inverse, and that turns solving the system into one multiplication.

Concept The three matrices

— the coefficient matrix, the numbers in front of the unknowns.

— the column of unknowns.

— the column of constants on the right-hand side.

Together they form , whatever the size of the system.

Theorem Multiply by the inverse

There is no division for matrices. Instead, multiply both sides on the left by :

Since

and

Order matters: matrix multiplication is not commutative, so must go on the left of both sides.

Example Step 1 — write it as AX = B

Take the system:

In matrix form:

Example Step 2 — find A⁻¹

The determinant:

Swap the main diagonal, flip the other signs:

Divide by the determinant 1, which changes nothing:

Example Step 3 — compute X = A⁻¹B

Each entry of the answer is a row of

multiplied against the column

⟹ x = 80 and y = 20

Example Step 4 — check the solution

Substitute both values back into the original equations.

✓

⟹ both equations hold, so the solution is correct

Note Points to watch

there is no inverse, so this method cannot be used.

Multiply by

on the left of both sides, never on the right.

Keep the equations in the same variable order when building

The method extends to larger systems, not just 2×2.

Summary

Every linear system can be written as AX = B.
There is no matrix division; multiply by A⁻¹ on the left instead.
This gives the solution rule X = A⁻¹B.
The worked system gave x = 80 and y = 20.
A⁻¹ exists only when det(A) ≠ 0; always substitute back to check.