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

Explanation

To find the inverse of the matrix \(\begin{pmatrix} 4 & 5 \\ 3 & 4 \end{pmatrix}\), we will use the method of row reduction to transform the matrix into the identity matrix while applying the same operations to the identity matrix.

### Step-by-Step Solution:

1. **Set up the augmented matrix**:
We start with the matrix \( A = \begin{pmatrix} 4 & 5 \\ 3 & 4 \end{pmatrix} \) and the identity matrix \( I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \). We form the augmented matrix \(\begin{pmatrix} 4 & 5 & | & 1 & 0 \\ 3 & 4 & | & 0 & 1 \end{pmatrix}\).

**Hint**: Write the augmented matrix with the original matrix on the left and the identity matrix on the right.

2. **Perform row operations to get a leading 1 in the first row**:
Divide the first row by 4:
\[ R_1 = \frac{1}{4} R_1 \Rightarrow \begin{pmatrix} 1 & \frac{5}{4} & | & \frac{1}{4} & 0 \\ 3 & 4 & | & 0 & 1 \end{pmatrix} \]

**Hint**: To create a leading 1, you can divide the entire row by the leading coefficient.

3. **Eliminate the first column of the second row**:
Replace the second row by subtracting 3 times the first row from it:
\[ R_2 = R_2 - 3R_1 \Rightarrow \begin{pmatrix} 1 & \frac{5}{4} & | & \frac{1}{4} & 0 \\ 0 & -\frac{1}{4} & | & -\frac{3}{4} & 1 \end{pmatrix} \]

**Hint**: Use row operations to eliminate the elements below the leading 1.

4. **Get a leading 1 in the second row**:
Multiply the second row by -4:
\[ R_2 = -4R_2 \Rightarrow \begin{pmatrix} 1 & \frac{5}{4} & | & \frac{1}{4} & 0 \\ 0 & 1 & | & 3 & -4 \end{pmatrix} \]

**Hint**: Normalize the second row to create a leading 1.

5. **Eliminate the second column of the first row**:
Replace the first row by subtracting \(\frac{5}{4}\) times the second row:
\[ R_1 = R_1 - \frac{5}{4}R_2 \Rightarrow \begin{pmatrix} 1 & 0 & | & -\frac{11}{4} & 5 \\ 0 & 1 & | & 3 & -4 \end{pmatrix} \]

**Hint**: Use row operations again to eliminate the second column in the first row.

6. **Final augmented matrix**:
The augmented matrix now looks like this:
\[ \begin{pmatrix} 1 & 0 & | & -\frac{11}{4} & 5 \\ 0 & 1 & | & 3 & -4 \end{pmatrix} \]
Thus, the inverse of the matrix \( A \) is:
\[ A^{-1} = \begin{pmatrix} -\frac{11}{4} & 5 \\ 3 & -4 \end{pmatrix} \]

### Final Answer:
The inverse of the matrix \(\begin{pmatrix} 4 & 5 \\ 3 & 4 \end{pmatrix}\) is:
\[ \begin{pmatrix} -\frac{11}{4} & 5 \\ 3 & -4 \end{pmatrix} \]