and what has been here proved with reference to the first column is equally true for any of the columns or rows; hence it appears that in reducing a determinant we may replace any one of the rows or columns by a new row or column formed in the following way:
Take the constituents of the row or column to be replaced, and increase or diminish them by any equimultiples of the corresponding constituents of one or more of the other rows or columns.
After a little practice it will be found that determinants may often be quickly simplified by replacing two or more rows or columns simultaneously: for example, it is easy to see that
but in any modification of the rule as above enunciated, care must be taken to leave one row or column unaltered.
Thus, if on the left-hand side of the last identity the constituents of the third column were replaced by respectively, we should have the former value increased by
and of the four determinants into which this may be re- solved there is one which does not vanish, namely
Ex. 1. Find the value of .