Solve the system $3x+4y=13$, $2x-y=1$ using the matrix inverse method.

Step-by-step solution

1. Write the coefficient matrix $C=\left(\begin{array}{cc}3& 4\\ 2& -1\end{array}\right)$ and the constant vector $\mathbf{b}=\left(\begin{array}{c}13\\ 1\end{array}\right)$.
2. Find the determinant of $C$: $\det C=3(-1)-4\cdot 2=-3-8=-11$.
3. Since $\det C\ne 0$, $C$ is invertible. Its inverse is $C^{-1}=\frac{1}{\det C}\left(\begin{array}{cc}-1& -4\\ -2& 3\end{array}\right)=\frac{1}{11}\left(\begin{array}{cc}1& 4\\ 2& -3\end{array}\right)$.
4. Multiply $C^{-1}$ by $\mathbf{b}$: $\mathbf{x}=C^{-1}\mathbf{b}=\frac{1}{11}\left(\begin{array}{cc}1& 4\\ 2& -3\end{array}\right)\left(\begin{array}{c}13\\ 1\end{array}\right)=\frac{1}{11}\left(\begin{array}{c}13+4\\ 26-3\end{array}\right)=\frac{1}{11}\left(\begin{array}{c}17\\ 23\end{array}\right)$.
5. Thus $x=\frac{17}{11}$ and $y=\frac{23}{11}$, i.e. $(x,y)=\left(\frac{17}{11},\frac{23}{11}\right)$.

Concept tested: Solving linear equations by matrix inverse

Published 19 September 2026 · How this solution was prepared and checked