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Problem 6: Finding All $2 \times 2$ Matrices That Commute with a Given Matrix

We need to find all $2 \times 2$ matrices $B = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ that commute with the matrix $A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}.$

For two matrices to commute, $AB = BA$. Therefore, we calculate both $AB$ and $BA$ and equate them.

1. Calculate $AB$: $AB = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 2a + 3c & 2b + 3d \\ a + 4c & b + 4d \end{pmatrix}.$

2. Calculate $BA$: $BA = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} = \begin{pmatrix} 2a + b & 3a + 4b \\ 2c + d & 3c + 4d \end{pmatrix}.$

3. Equate $AB = BA$: $\begin{pmatrix} 2a + 3c & 2b + 3d \\ a + 4c & b + 4d \end{pmatrix} = \begin{pmatrix} 2a + b & 3a + 4b \\ 2c + d & 3c + 4d \end{pmatrix}.$

4. Set up the system of equations by comparing elements:

   • From the (1,1) position: $2a + 3c = 2a + b \Rightarrow b = 3c$.
   • From the (1,2) position: $2b + 3d = 3a + 4b \Rightarrow 3d = 3a + 2b$.
   • From the (2,1) position: ( a + 4c = 2c + d \Rightarrow d = -