Find the inverse of the following , if it exists, by using elementary row (column) transformations :
`[(6,-3),(-2,1)]`

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} 6 & -3 \\ -2 & 1 \end{pmatrix} \) using elementary row transformations, we will follow these steps:

### Step 1: Write the augmented matrix
We start by writing the augmented matrix \([A | I]\), where \(I\) is the identity matrix of the same size as \(A\).

\[ \begin{pmatrix} 6 & -3 & | & 1 & 0 \\ -2 & 1 & | & 0 & 1 \end{pmatrix} \]

### Step 2: Find the determinant of \(A\)
To determine if the inverse exists, we need to calculate the determinant of \(A\).

\[ \text{det}(A) = (6)(1) - (-3)(-2) = 6 - 6 = 0 \]

### Step 3: Analyze the determinant
Since the determinant of \(A\) is \(0\), it indicates that the matrix is singular, and therefore, it does not have an inverse.

### Final Conclusion
The inverse of the matrix \( A \) does not exist.

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