To find the inverse of a given $2 \times 2$ matrix using row reduction, we follow these steps. The given matrix is:

$A = \begin{pmatrix} 4 & -1 \\ -12 & 3 \end{pmatrix}$

We augment this matrix with the identity matrix:

$\left( \begin{array}{cc|cc} 4 & -1 & 1 & 0 \\ -12 & 3 & 0 & 1 \end{array} \right)$

Now we will use Gaussian elimination (row reduction) to reduce the left side of this augmented matrix to the identity matrix, and the right side will transform into the inverse of the original matrix.

Step 1: Make the top-left element a 1

We divide the first row by 4:

$\left( \begin{array}{cc|cc} 1 & -\frac{1}{4} & \frac{1}{4} & 0 \\ -12 & 3 & 0 & 1 \end{array} \right)$

Step 2: Eliminate the first entry of the second row

We add 12 times the first row to the second row to eliminate the $-12$ in the second row, first column:

$\left( \begin{array}{cc|cc} 1 & -\frac{1}{4} & \frac{1}{4} & 0 \\ 0 & 0 & 3 & 1 \end{array} \right)$

Step 3: Check for row issues

At this point, the second row has all zeroes on the left side, meaning that the determinant of the matrix is 0. A matrix with determinant 0 is singular, meaning it does not have an inverse.

Thus, the inverse of this matrix does not exist (DNE).

Would you like more details on any of these steps?

Related Questions:

1. How do you determine if a matrix is singular?
2. Can row reduction be used to find the inverse of a larger matrix?
3. What does the determinant tell us about the invertibility of a matrix?
4. How is the determinant of a $2 \times 2$ matrix calculated?
5. What are the properties of non-invertible matrices?

Tip: When checking if a matrix is invertible, calculating the determinant first can save time.