If `A=[{:(2,3),(5,-2):}]`, write `A^(-1)` in terms of A.

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} 2 & 3 \\ 5 & -2 \end{pmatrix} \) in terms of \( A \), we will follow these steps:

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

The determinant of a \( 2 \times 2 \) matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated using the formula:

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

For our matrix \( A \):

- \( a = 2 \)
- \( b = 3 \)
- \( c = 5 \)
- \( d = -2 \)

Calculating the determinant:

\[ \text{det}(A) = (2)(-2) - (3)(5) = -4 - 15 = -19 \]

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

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

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

For our matrix \( A \):

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

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

The inverse of \( A \) is given by the formula:

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

Substituting the values we found:

\[ A^{-1} = \frac{1}{-19} \begin{pmatrix} -2 & -3 \\ -5 & 2 \end{pmatrix} \]

This simplifies to:

\[ A^{-1} = \begin{pmatrix} \frac{2}{19} & \frac{3}{19} \\ \frac{5}{19} & -\frac{2}{19} \end{pmatrix} \]

### Step 4: Express \( A^{-1} \) in terms of \( A \)

Notice that we can factor out \( -\frac{1}{19} \) from the adjoint matrix:

\[ A^{-1} = -\frac{1}{19} \begin{pmatrix} 2 & 3 \\ 5 & -2 \end{pmatrix} = -\frac{1}{19} A \]

Thus, we can express \( A^{-1} \) in terms of \( A \):

\[ A^{-1} = -\frac{1}{19} A \]

### Final Result

\[ A^{-1} = -\frac{1}{19} A \]

---