Linear systems in matrix form

Matrix notation lets us write a whole system on a single line and solve it with the inverse.

The system $\begin{cases} a_1x+b_1y=c_1 \\ a_2x+b_2y=c_2\end{cases}$ is rewritten as $AX=C$ with

$A = \begin{pmatrix} a_1 & b_1 \\ a_2 & b_2\end{pmatrix},\quad X = \begin{pmatrix} x \\ y\end{pmatrix},\quad C = \begin{pmatrix} c_1 \\ c_2\end{pmatrix}.$

If $\det A\ne 0$ we have $X = A^{-1}C$. It is the same result as Cramer’s, written in compact notation.

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Topics: Linear systems
Concepts: Matrix form · Inverse matrix · Cramer’s rule
Skills: Solve systems