We are given the equation:

$\det(X^2 - 5I_2) = 0$

where $X$ is a $2 \times 2$ matrix and $I_2$ is the $2 \times 2$ identity matrix. We need to prove that this implies $X^2 = 5I_2$.

Step-by-step Proof:

1. Use of the Determinant Property:
   The determinant of a matrix being 0 implies that the matrix itself is singular, i.e., it has no inverse. In this case, it means that: $\det(X^2 - 5I_2) = 0$ implies that $X^2 - 5I_2$ is a singular matrix.

2. Eigenvalues and Determinants:
   Let’s assume $\lambda$ is an eigenvalue of the matrix $X$. For any matrix $X$, its eigenvalues satisfy the equation $\det(X - \lambda I) = 0$.

   Now, since $X^2 - 5I_2$ is singular, the eigenvalues of $X^2 - 5I_2$ must include 0. That is, for eigenvalues $\lambda$ of $X$, we have: $\lambda^2 - 5 = 0$ or $\lambda^2 = 5$ which means the eigenvalues of $X$ are $\lambda = \pm \sqrt{5}$.

3. Diagonalization:
   Since $X$ is a $2 \times 2$ matrix and has two distinct eigenvalues ($\pm \sqrt{5}$), it can be diagonalized. That is, there exists an invertible matrix $P$ and a diagonal matrix $D$ such that: $X = P D P^{-1}$ where $D = \text{diag}(\sqrt{5}, -\sqrt{5})$.

4. Square of $X$:
   Squaring both sides of $X = P D P^{-1}$: $X^2 = P D^2 P^{-1}$ Since $D^2 = \text{diag}(5, 5) = 5I_2$, we have: $X^2 = P (5I_2) P^{-1} = 5P I_2 P^{-1} = 5I_2$

Thus, we have shown that $X^2 = 5I_2$.

Conclusion:

The fact that $\det(X^2 - 5I_2) = 0$ implies that the eigenvalues of $X$ are $\pm \sqrt{5}$, and hence $X^2 = 5I_2$, as required.

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Do you want further details on any of the steps, or would you like more examples?

Here are five related questions you might consider:

1. How do eigenvalues affect the determinant of a matrix?
2. Can you prove a similar result for a $3 \times 3$ matrix?
3. How does diagonalization help in simplifying matrix problems?
4. Can the proof work without assuming diagonalizability of $X$?
5. What would happen if $X$ had repeated eigenvalues?

Tip: Always check if a matrix is diagonalizable when solving matrix equations. Diagonalization simplifies the process significantly!