### Instruction: Find the inverse of the matrix
\[\begin{pmatrix} 2 & 3 \\ -1 & 7 \end{pmatrix}.\]If the inverse does not exist, then enter the zero matrix.### Response: To find the inverse of a 2 x 2 matrix, I can use the formula
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix},\]
as long as the determinant $ad-bc$ is not zero.
In this case, the determinant is $2 \times 7 - 3 \times (-1) = 14 + 3 = 17$, which is nonzero, so the inverse does exist.
Applying the formula, I get
\[\begin{pmatrix} 2 & 3 \\ -1 & 7 \end{pmatrix}^{-1} = \frac{1}{17}\begin{pmatrix} 7 & -3 \\ 1 & 2 \end{pmatrix}.\]