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 formulas related to 2x2 matrices include the determinant and the inverse. 4. The determinant is given by: $$\det = ad - bc$$ 5. The inverse of the matrix (if \(\det \neq 0\)) is: $$\begin{bmatrix}a & b \\ c & d\end{bmatrix}^{-1} = \frac{1}{ad - bc} \begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$$ 6. The determinant tells us if the matrix is invertible: if \(ad - bc \neq 0\), the matrix has an inverse. 7. This matrix can represent linear transformations in 2D space, such as rotations, scalings, and shears. 8. Understanding these basics helps in solving systems of linear equations, finding eigenvalues, and more.