1. The Shift: Subtract $\lambda$ from the top-left and bottom-right entries.
2. The Determinant: Multiply the diagonals: $(a-\lambda)(d-\lambda) - bc$.
3. Expand: Use FOIL to multiply the binomials.
4. Solve: Set the quadratic to zero and find the values of $\lambda$.
Find the characteristic polynomial ($\\lambda^2 + b\\lambda + c$) for each matrix.
$A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$
$B = \begin{bmatrix} 5 & 0 \\ 0 & 2 \end{bmatrix}$ (A Diagonal Matrix).
$C = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$
Solve the quadratic equations you found in Part I to find the eigenvalues ($\\lambda_1, \\lambda_2$).
For Matrix A: $\\lambda^2 - 6\\lambda + 5 = 0$
For Matrix B: (Solve your polynomial from Part I).
For Matrix C: (Solve your polynomial from Part I).
In Matrix A, add the two eigenvalues ($1+5 = 6$). Now add the main diagonal of the original matrix ($2+4 = 6$). Do they match? Explain why the "Sum of Life" is preserved in the DNA.
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Find the eigenvalues of the Identity Matrix $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.
Find the characteristic equation for the 90-degree Rotation Matrix $R = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$.
Task: Set up $\\det(R - \\lambda I) = 0$ and simplify.
(Hint: Your answer will be $\\lambda^2 + 1 = 0$. Can you solve this with Real numbers? What does this tell you about the DNA of a rotation?)
Objective: Explain the "Characteristic" to a younger sibling using a packet of flower seeds.
The Activity:
1. Show them the seeds.
2. Ask: "Can you tell what color the flowers will be just by looking at the seeds?"
3. "No, but the instruction is hidden inside. In math, we have a way to find those hidden instructions."
The Lesson: "Our identity is like a seed. God has hidden His plan for our beauty inside us, even before we start growing."
Response: ___________________________________________________________