Find the inverse of the following matrices if exist :
(i) `|{:(5,-3),(2,2):}|`
(ii) `|{:(1,-3),(-1,2):}|`
(iii) `|{:(1,0,-1),(3,4,5),(0,-6,-7):}|`
(iv) `|{:(1,-3,3),(2,2,-4),(2,0,2):}|`
(v) `|{:(1,2,1),(1,-1,-2),(1,2,-1):}|`
(vi) `|{:(4,-2,-1),(1,1,-1),(-1,2,4):}|`

Answer

To find the inverse of the given matrices, we will follow a systematic approach. The inverse of a matrix \( A \) exists if and only if the determinant of \( A \) is non-zero. The formula for the inverse of a \( 2 \times 2 \) matrix is given by:

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

For a \( 3 \times 3 \) matrix, the inverse can be found using the formula:

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

where \( \text{adj}(A) \) is the adjoint of \( A \), which is the transpose of the cofactor matrix.

### (i) Matrix:
\[ A = \begin{pmatrix} 5 & -3 \\ 2 & 2 \end{pmatrix} \]

**Step 1: Calculate the determinant.**
\[ \text{det}(A) = 5 \cdot 2 - (-3) \cdot 2 = 10 + 6 = 16 \]

**Step 2: Check if the determinant is non-zero.**
Since \( \text{det}(A) = 16 \neq 0 \), the inverse exists.

**Step 3: Calculate the adjoint.**
For a \( 2 \times 2 \) matrix, the adjoint is given by:
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \begin{pmatrix} 2 & 3 \\ -2 & 5 \end{pmatrix} \]

**Step 4: Calculate the inverse.**
\[ A^{-1} = \frac{1}{16} \begin{pmatrix} 2 & 3 \\ -2 & 5 \end{pmatrix} = \begin{pmatrix} \frac{1}{8} & \frac{3}{16} \\ -\frac{1}{8} & \frac{5}{16} \end{pmatrix} \]