There are several ways to calculate the fourth-order determinant, and one of the commonly used methods is the benchmark second-order sub-formula method. This method has strong regularity, and through repeated application of this method, the fourth-order determinant can be gradually transformed into the third-order, and then from the third-order to the second-order, and finally the result can be easily calculated.
Another strategy is the elementary transformation method, which uses the elementary transformation of the determinant to transform the original determinant into an easily evaluated determinant (e.g., the triangular determinant). There are three elementary transformations of the determinant: swapping any two lines of the determinant, and changing the determinant sign;Multiply a row by a non-zero constant k, and the value of the determinant becomes k times the original;Add a constant multiple of one row to another, and the determinant value does not change.
In addition, there is a strategy known as the "addition method", which expands the fourth-order determinant to the fifth-order determinant, and then eliminates the second row and the elements following it by adding the opposite number of the elements in the first row to each element in addition to the first row1.
The above is a common way to calculate the fourth-order determinant, and you can choose the most suitable calculation strategy for your specific problem and needs.