1. **State the problem:** We are given a matrix $$H = \begin{bmatrix} 1 & -3 \\ 2 & 4 \end{bmatrix}$$ and need to find $$H^2 = H \times H$$. 2. **Recall matrix multiplication:** To multiply two 2x2 matrices $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ and $$B = \begin{bmatrix} e & f \\ g & h \end{bmatrix}$$, the product $$AB$$ is: $$AB = \begin{bmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{bmatrix}$$ 3. **Apply the formula to $$H \times H$$:** Let $$H = \begin{bmatrix} 1 & -3 \\ 2 & 4 \end{bmatrix}$$, then $$H^2 = \begin{bmatrix} 1 & -3 \\ 2 & 4 \end{bmatrix} \times \begin{bmatrix} 1 & -3 \\ 2 & 4 \end{bmatrix}$$ Calculate each element: - Top-left element: $$1 \times 1 + (-3) \times 2 = 1 - 6 = -5$$ - Top-right element: $$1 \times (-3) + (-3) \times 4 = -3 - 12 = -15$$ - Bottom-left element: $$2 \times 1 + 4 \times 2 = 2 + 8 = 10$$ - Bottom-right element: $$2 \times (-3) + 4 \times 4 = -6 + 16 = 10$$ 4. **Write the final matrix:** $$H^2 = \begin{bmatrix} -5 & -15 \\ 10 & 10 \end{bmatrix}$$ This is the product of matrix $$H$$ with itself.