1. The problem shows a matrix $J$ defined with trigonometric expressions: $$J = \begin{pmatrix}-2 \times 4 \times \sin(4) & \cos(4) - 4 \times \sin(4) \\ 2 \times \cos(2) + 1 & 1 \times \cos(2) - 1\end{pmatrix}$$ 2. We are asked to understand what this matrix represents and verify the calculation. 3. First, calculate the elements symbolically: - Top-left: $-2 \times 4 \times \sin(4) = -8 \sin(4)$ - Top-right: $\cos(4) - 4 \sin(4)$ - Bottom-left: $2 \cos(2) + 1$ - Bottom-right: $1 \cos(2) - 1 = \cos(2) -1$ 4. Using approximate values for sine and cosine: - $\sin(4) \approx -0.7568$ - $\cos(4) \approx -0.6536$ - $\cos(2) \approx -0.4161$ 5. Substitute these values: - Top-left: $-8 \times (-0.7568) = 6.0544$ - Top-right: $-0.6536 - 4 \times (-0.7568) = -0.6536 + 3.0272 = 2.3736$ - Bottom-left: $2 \times (-0.4161) + 1 = -0.8322 + 1 = 0.1678$ - Bottom-right: $-0.4161 - 1 = -0.5839$ 6. So, the matrix is: $$J = \begin{pmatrix}6.0544 & 2.3736 \\ 0.1678 & -0.5839\end{pmatrix}$$ 7. Interpretation: This matrix looks like a Jacobian matrix where each element is a partial derivative evaluated numerically at certain points (likely $x=2$, $y=4$) in some function involving sines and cosines. Final answer: The given matrix $J$ is a numerical Jacobian matrix with entries computed from trigonometric functions, displayed and evaluated numerically.