`[{:(4,5),(3,4):}]` find the inverse of matrix

Explanation

To find the inverse of the matrix \( A = \begin{pmatrix} 4 & 5 \\ 3 & 4 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of Matrix A
The determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix:
- \( a = 4 \)
- \( b = 5 \)
- \( c = 3 \)
- \( d = 4 \)

Calculating the determinant:
\[ \text{det}(A) = (4)(4) - (5)(3) = 16 - 15 = 1 \]

### Step 2: Calculate the Adjoint of Matrix A
The adjoint of a 2x2 matrix is found by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. The formula for the adjoint is:
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
For our matrix:
- \( d = 4 \)
- \( b = 5 \)
- \( c = 3 \)
- \( a = 4 \)

Calculating the adjoint:
\[ \text{adj}(A) = \begin{pmatrix} 4 & -5 \\ -3 & 4 \end{pmatrix} \]

### Step 3: Calculate the Inverse of Matrix A
The inverse of a matrix \( A \) is given by the formula:
\[ A^{-1} = \frac{\text{adj}(A)}{\text{det}(A)} \]
Substituting the values we calculated:
\[ A^{-1} = \frac{1}{1} \begin{pmatrix} 4 & -5 \\ -3 & 4 \end{pmatrix} = \begin{pmatrix} 4 & -5 \\ -3 & 4 \end{pmatrix} \]

### Final Answer
Thus, the inverse of the matrix \( A \) is:
\[ A^{-1} = \begin{pmatrix} 4 & -5 \\ -3 & 4 \end{pmatrix} \]

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