If `A = [(3,1),(5,2)]`, and AB = BA = I, then find the matrix B.

Answer

To find the matrix \( B \) such that \( AB = BA = I \), where \( A = \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix} \), we need to calculate the inverse of matrix \( A \).

### Step 1: Calculate the Determinant of \( A \)

The determinant of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:

\[ \text{det}(A) = ad - bc \]

For our matrix \( A \):

\[ \text{det}(A) = (3)(2) - (5)(1) = 6 - 5 = 1 \]

### Step 2: Find the Adjoint of \( A \)

The adjoint of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:

\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

For our matrix \( A \):

\[ \text{adj}(A) = \begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix} \]

### Step 3: Calculate the Inverse of \( A \)

The inverse of a matrix \( A \) is given by:

\[ A^{-1} = \frac{\text{adj}(A)}{\text{det}(A)} \]

Since we calculated that \( \text{det}(A) = 1 \):

\[ A^{-1} = \frac{1}{1} \begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix} = \begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix} \]

### Step 4: Conclude the Result

Since \( B = A^{-1} \), we have:

\[ B = \begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix} \]

Thus, the matrix \( B \) is:

\[ \boxed{\begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix}} \]

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