Determine the minor of $a_{22}$ in the matrix. \[ A=\left[\begin{array}{ccc} 2 & -4 & 5 \\ 2 & 5 & 2 \\ 3 & 7 & -6 \end{array}\right] \]

Step 1 ：Given the matrix A = \(\left[\begin{array}{ccc} 2 & -4 & 5 \\ 2 & 5 & 2 \\ 3 & 7 & -6 \end{array}\right]\)

Step 2 ：We need to find the minor of the element at the second row and second column, i.e., 5.

Step 3 ：To do this, we delete the second row and second column from the matrix A. The resulting sub-matrix is \(\left[\begin{array}{cc} 2 & 5 \\ 3 & -6 \end{array}\right]\)

Step 4 ：The minor of the element is the determinant of this sub-matrix.

Step 5 ：The determinant of a 2x2 matrix \(\left[\begin{array}{cc} a & b \\ c & d \end{array}\right]\) is given by \(ad - bc\).

Step 6 ：Applying this to our sub-matrix, we get \((2*-6) - (5*3) = -27\)

Step 7 ：Final Answer: The minor of the element at the second row and second column in the matrix A is \(\boxed{-27}\)