If the determinant of X^2-5I2 is 0, prove that X^2=5I2

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.

---

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!

If the determinant of X^2 - 5I2 is 0, prove that X^2 = 5I2

Learn how to prove that X^2 = 5I2 by analyzing the determinant of a matrix and applying concepts like eigenvalues and diagonalization. This step-by-step solution is ideal for undergraduate linear algebra students.

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