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439 Rh 550. A Determinant expressed as the Sum of Two Other Determinants.

If each constituent in any row, or column, consists of two terms, then the determinant can be expressed as the sum of two other determinants.

Thus we have

for the expression on the left,

which proves the proposition.

In like manner if each constituent in any one row, or column, consists of m terms, the determinant can be expressed as the sum of m other determinants.

Similarly, we may show that

In general if the constituents of the three columns consist of m, n, p terms, respectively, the determinant can be expressed as the sum of mnp determinants.

Ex. 1. Show that

The given determinant