1. The problem is to find the product of the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\) with itself, i.e., compute \(\begin{bmatrix}a & b \\ c & d\end{bmatrix} \times \begin{bmatrix}a & b \\ c & d\end{bmatrix}\). 2. The formula for matrix multiplication of two 2x2 matrices \(\begin{bmatrix}p & q \\ r & s\end{bmatrix}\) and \(\begin{bmatrix}w & x \\ y & z\end{bmatrix}\) is: $$\begin{bmatrix}p & q \\ r & s\end{bmatrix} \times \begin{bmatrix}w & x \\ y & z\end{bmatrix} = \begin{bmatrix}pw + qy & px + qz \\ rw + sy & rx + sz\end{bmatrix}$$ 3. Applying this to our matrix: - Top-left element: \(a \times a + b \times c = a^2 + bc\) - Top-right element: \(a \times b + b \times d = ab + bd\) - Bottom-left element: \(c \times a + d \times c = ca + dc\) - Bottom-right element: \(c \times b + d \times d = cb + d^2\) 4. Therefore, the product matrix is: $$\begin{bmatrix}a^2 + bc & ab + bd \\ ca + dc & cb + d^2\end{bmatrix}$$ This is the final answer.