1. Solve the Quadratic: $\lambda^2 - (\text{Trace } A)\lambda + \det A = 0$.
2. Factor or Formula: Use $(\lambda - \lambda_1)(\lambda - \lambda_2) = 0$ to find the roots.
3. Check the Witnesses: The sum must be the Trace, and the product must be the Determinant.
4. Diagonal Rule: For diagonal matrices, the eigenvalues are already on the diagonal!
Find the two eigenvalues ($\lambda_1, \lambda_2$) for each matrix.
$A = \begin{bmatrix} 2 & 3 \\ 0 & 5 \end{bmatrix}$
$B = \begin{bmatrix} 3 & 4 \\ 4 & 3 \end{bmatrix}$
$C = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$
For Matrix C above, what are the two eigenvalues? What is their product? Why is the determinant of this matrix zero? Explain why this matrix represents a "Singular" life that collapses in one direction.
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Based on the eigenvalues, decide if the transformation is **Expansion** ($>1$), **Contraction** ($<1$), or **Inversion** ($<0$).
$\lambda = 5$
$\lambda = -2$
$\lambda = 0.1$
Find the eigenvalues of $A = \begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix}$.
1. Trace = ...
2. Det = ...
3. Equation: $\lambda^2 + 4 = 0$.
Task: Solve for $\lambda$. (Hint: $\lambda = \sqrt{-4}$).
Objective: Explain Eigenvalues to a younger sibling using a magnifying glass.
The Activity:
1. Look at an ant through the glass. "It looks 10 times bigger! The Eigenvalue is 10."
2. Look at the ant through the wrong end of binoculars. "It looks tiny! The Eigenvalue is 0.1."
The Lesson: "God has a magnifying glass for our lives. Some things He makes huge, and some things He keeps small. Both are part of His plan."
Response: ___________________________________________________________