1. The problem is to understand the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\) and its properties. 2. This is a 2x2 matrix with elements \(a, b, c, d\). 3. One important property is the determinant, calculated by the formula: $$\text{det} = ad - bc$$ 4. The determinant tells us if the matrix is invertible (non-zero determinant) or singular (zero determinant). 5. Another property is the trace, which is the sum of the diagonal elements: $$\text{trace} = a + d$$ 6. These properties are fundamental in linear algebra for solving systems of equations, finding eigenvalues, and more. Final answer: The matrix is \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\) with determinant \(ad - bc\) and trace \(a + d\).