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. Important operations include finding the determinant, inverse, and eigenvalues. 4. The determinant formula for a 2x2 matrix is: $$\det = ad - bc$$ 5. The inverse exists only if \(\det \neq 0\), and is given by: $$\begin{bmatrix}a & b \\ c & d\end{bmatrix}^{-1} = \frac{1}{ad - bc} \begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$$ 6. Eigenvalues \(\lambda\) satisfy: $$\det\left(\begin{bmatrix}a & b \\ c & d\end{bmatrix} - \lambda \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\right) = 0$$ which simplifies to: $$\det\begin{bmatrix}a - \lambda & b \\ c & d - \lambda\end{bmatrix} = 0$$ 7. Expanding the determinant: $$ (a - \lambda)(d - \lambda) - bc = 0 $$ 8. This quadratic equation in \(\lambda\) can be solved to find eigenvalues. This explanation covers the basic understanding and key formulas related to the given matrix.