Solving the Adjugate of A Cubed Minus B Cubed

Animation Specification: Solution to |adj(A^3 - B^3)|

Animation Description and Purpose

This animation explains the step-by-step solution to finding the absolute value of the adjugate of $A^3 - B^3$ for given matrices $A$ and $B$. The animation will visually derive the matrices, compute $A^3 - B^3$, and show the final determinant calculation.

Mathematical Elements and Formulas

1. Matrix $A$: Initially shown as $A = \begin{bmatrix} d & 2 \\ 1 & 2 \end{bmatrix}$, with the equation $A^2 - 4A + 2I = 0$. The diagonal elements $(d, 2)$ are highlighted, and the equation $d + 2 = 4$ is derived to show $d = 2$.
2. Matrix $B$: Shown as $B = \begin{bmatrix} 1 & 1 \\ \beta & 1 \end{bmatrix}$, with the equation $B^2 - 2B - I = 0$. The determinant $(1 - \beta)$ is highlighted, and the equation $1 - \beta = -1$ is derived to show $\beta = 2$.
3. Final Matrices: Display $A = \begin{bmatrix} 2 & 2 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 1 \\ 2 & 1 \end{bmatrix}$.
4. Matrix $M$: Compute $M = A^3 - B^3$, resulting in $M = \begin{bmatrix} 13 & 23 \\ 4 & 13 \end{bmatrix}$.
5. Final Step: For a $2 \ imes 2$ matrix, $|\ ext{adj}(M)| = |M|$. Show the calculation $(13 \ imes 13) - (23 \ imes 4) = 169 - 92 = 77$. Highlight the final answer $77$ in yellow.

Visual Elements

1. Matrices: Display matrices $A$ and $B$ with clear, bold text. Use a light gray background for matrices to distinguish them from the background.
2. Equations: Show equations $A^2 - 4A + 2I = 0$ and $B^2 - 2B - I = 0$ with a white background and black text for readability.
3. Highlights: Use a yellow highlight for the diagonal elements $(d, 2)$ in $A$ and the determinant $(1 - \beta)$ in $B$. Highlight the final answer $77$ in yellow.
4. Transitions: Smooth transitions between steps, with matrices and equations fading in and out as needed.

Animation Timing and Transitions

1. Introduction (0-3 seconds): Display the initial matrices $A$ and $B$ with their respective equations.
2. Derivation of $d$ (3-6 seconds): Highlight the diagonal elements $(d, 2)$ and show the derivation $d + 2 = 4$, leading to $d = 2$.
3. Derivation of $\beta$ (6-9 seconds): Highlight the determinant $(1 - \beta)$ and show the derivation $1 - \beta = -1$, leading to $\beta = 2$.
4. Final Matrices (9-12 seconds): Display the final matrices $A$ and $B$ with their computed values.
5. Computation of $M$ (12-18 seconds): Show the calculation of $M = A^3 - B^3$, resulting in $M = \begin{bmatrix} 13 & 23 \\ 4 & 13 \end{bmatrix}$.
6. Final Step (18-24 seconds): Show the calculation $(13 \ imes 13) - (23 \ imes 4) = 169 - 92 = 77$, highlighting the final answer $77$ in yellow.

Camera Angles and Perspectives

• Use a static camera angle centered on the matrices and equations. Zoom in slightly on the highlighted elements to emphasize their importance.

Additional Details

• Text Display: Use an opaque white background for all text to ensure readability. Avoid overlapping text with other elements.
• Color Scheme: Use a light gray background for matrices, white background for equations, and yellow for highlights. Use black text for all elements to ensure contrast.
• Animation Style: Use smooth transitions and fading effects to maintain clarity and focus on the current step.