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Let's solve these problems one by one.

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### Problem 6
**Find all \( 2 \times 2 \) matrices that commute with the matrix**
\[
\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}.
\]

To find all matrices \( B = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) that commute with \( A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \), we need:
\[
AB = BA.
\]

Expanding both \( AB \) and \( BA \) and equating the entries will yield a system of equations in terms of \( a \), \( b \), \( c \), and \( d \).

1. Compute \( AB \):
   \[
   AB = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} 2a + 3c & 2b + 3d \\ a + 4c & b + 4d \end{bmatrix}.
   \]

2. Compute \( BA \):
   \[
   BA = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} = \begin{bmatrix} 2a + b & 3a + 4b \\ 2c + d & 3c + 4d \end{bmatrix}.
   \]

Equating \( AB \) and \( BA \):
\[
\begin{cases}
2a + 3c = 2a + b, \\
2b + 3d = 3a + 4b, \\
a + 4c = 2c + d, \\
b + 4d = 3c + 4d.
\end{cases}
\]

Solving this system will give the set of matrices that commute with \( A \).

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### Problem 7
**If \( A = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix} \), show that \( A^2 - 2A + I_2 = O_2 \). Hence, find \( A^{50} \).**

1. **Calculate \( A^2 \):**
   \[
   A^2 = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix} = A.
   \]

2. **Substitute into \( A^2 - 2A + I_2 \):**
   \[
   A^2 - 2A + I_2 = A - 2A + I_2 = -A + I_2 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = O_2.
   \]

This shows that \( A \) satisfies the equation \( A^2 - 2A + I_2 = O_2 \), meaning \( A \) is idempotent.

3. **Find \( A^{50} \):**
   Since \( A = A^2 \), \( A^n = A \) for any \( n \geq 1 \). Therefore,
   \[
   A^{50} = A.
   \]

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Would you like further explanations or details on any part of this solution?

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#### Further Questions:
1. What are the implications of a matrix commuting with another?
2. How do idempotent matrices behave under matrix powers?
3. Can we generalize the solution for other matrices that satisfy similar properties?
4. How would you solve similar problems for \( 3 \times 3 \) matrices?
5. What happens if we modify the entries in \( A \) slightly? Does \( A^{50} \) change?

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#### Tip:
When working with matrix equations, it’s helpful to check for idempotent properties or symmetries, as these can simplify high powers significantly.

6. Find all 2 x 2 matrices which commute with the matrix [ [2, 3], [1, 4] ].
7. If A = [ [1, 0], [-1, 1] ], show that A^2 - 2A + I_2 = O_2. Hence find A^50.

Explore how to find all 2x2 matrices that commute with a given matrix and show that a matrix satisfies a specific quadratic identity. Additionally, learn how to calculate high powers of matrices by using idempotent properties. Suitable for college-level linear algebra.

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