COR. If each constituent of one row, or column, is the same multiple of the corresponding constituent of another row, or column, the determinant vanishes.

495. 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.

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.

These results may easily be generalised; thus 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.

Of these four determinants the first three vanish, Art. 493; thus the expression reduces to the last of the four determinants; hence its value

= - {c (c2 — ab) − b (ac − b2) + a (a2 – bc)}

=3abc-a3 - b3 — c3.

as in the last article we can shew that it is equal to

and the last two of these determinants vanish [Art. 494 Cor.]. Thus we see that the given determinant is equal to a new one whose first column is obtained by subtracting from the constituents of the first column of the original determinant equimultiples of the corresponding constituents of the other columns, while the second and third columns remain unaltered.

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 ΟΥ 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

that

see

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 c1+ra1, c2+ra, c2+ra, respectively, we should have the former value increased by

and of the four determinants into which this may be resolved there is one which does not vanish, namely

[Explanation. In the first step of the reduction keep the second column unaltered; for the first new column diminish each constituent of the first column by the corresponding constituent of the second; for the third new column diminish each constituent of the third column by the corresponding constituent of the second. In the second step take out the factors 3 and - 4. In the third step keep the first row unaltered; for the second new row diminish the constituents of the second by the corresponding ones of the first; for the third new row diminish the constituents of the third by twice the corresponding constituents of the first. The remaining steps will be easily seen.]

[Explanation. In the first new determinant the first row is the sum of the constituents of the three rows of the original determinant, the second and third rows being unaltered. In the third of the new determinants the first column remains unaltered, while the second and third columns are obtained by subtracting the constituents of the first column from those of the second and third respectively. The remaining transformations are sufficiently obvious.]

497. Before shewing how to express the product of two determinants as a determinant, we shall investigate the value of

From Art. 495, we know that the above determinant can be expressed as the sum of 27 determinants, of which it will be sufficient to give the following specimens:

the first of which vanishes; similarly it will be found that 21 out of the 27 determinants vanish. The six determinants that remain are equal to

hence the given determinant can be expressed as the product of two other determinants.

498. The product of two determinants is a determinant.

Consider the two linear equations