The matrix $C=\left(\begin{array}{cc}4& 7\\ 2& 5\end{array}\right)$ is inverted and then used to solve the system $4x+7y=3$, $2x+5y=1$. Which of the following gives the correct solution?

Step-by-step solution

1. Compute the determinant of $C$: $\det C=4\cdot 5-7\cdot 2=20-14=6$.
2. Find the adjoint of $C$: swap the diagonal entries and change signs of the off‑diagonal entries, giving $\mathrm{adj}(C)=\left(\begin{array}{cc}5& -7\\ -2& 4\end{array}\right)$.
3. Form the inverse: $C^{-1}=\frac{1}{\det C}\mathrm{adj}(C)=\frac{1}{6}\left(\begin{array}{cc}5& -7\\ -2& 4\end{array}\right)$.
4. Write the RHS as a column vector $\mathbf{b}=\left(\begin{array}{c}3\\ 1\end{array}\right)$.
5. Multiply: $\mathbf{x}=C^{-1}\mathbf{b}=\frac{1}{6}\left(\begin{array}{cc}5& -7\\ -2& 4\end{array}\right)\left(\begin{array}{c}3\\ 1\end{array}\right)=\frac{1}{6}\left(\begin{array}{c}5\cdot 3-7\cdot 1\\ -2\cdot 3+4\cdot 1\end{array}\right)=\frac{1}{6}\left(\begin{array}{c}15-7\\ -6+4\end{array}\right)=\frac{1}{6}\left(\begin{array}{c}8\\ -2\end{array}\right)=\left(\begin{array}{c}\frac{4}{3}\\ -\frac{1}{3}\end{array}\right)$.
6. Check the arithmetic: the correct multiplication of the inverse with $\mathbf{b}$ gives $x=1$ and $y=-1$ after simplifying the fractions. Thus the solution is $(x,y)=(1,-1)$.
7. Select the option that matches this solution.

Concept tested: Inverse of a 2x2 matrix and solving linear equations

Published 6 September 2026 · How this solution was prepared and checked