Theory of Equation

Determinants

554 Definitions: The determinant 𝑎₁𝑎₂𝑏₁𝑏₂ is equivalent to 𝑎₁𝑏₂−𝑎₂𝑏₁, and is called a determinant of the second order.
A determinant of the third order is 𝑎₁𝑎₂𝑎₃𝑏₁𝑏₂𝑏₃𝑐₁𝑐₂𝑐₃≡𝑎₁(𝑏₂𝑐₃−𝑏₃𝑐₂)+𝑎₂(𝑏₃𝑐₁−𝑏₁𝑐₃)+𝑎₃(𝑏₁𝑐₂−𝑏₂𝑐₁) Another notation is ∑±𝑎₁𝑏₂𝑐₃, or simply (𝑎₁𝑏₂𝑐₃).
The letters are named constituents, and the terms are called elements. The determinant is composed of all the elements obtained by permutations of the suffixes 1, 2, 3.
The coefficients of the constituents are determinants of the next lower order, and are termed minors of the original determinant. Thus, the first determinant above is the minor of 𝑐₃ in the second determinant. It is denoted by 𝐶₃. So the minor of 𝑎₁ is denoted by 𝐴₁, and so on. 555 A determinant of the 𝑛^th order may be written in either of the forms below 𝑎₁𝑎₂⋯𝑎_𝑟⋯𝑎_𝑛𝑏₁𝑏₂⋯𝑏_𝑟⋯𝑏_𝑛⋯⋯⋯⋯⋯⋯𝑙₁𝑙₂⋯𝑙_𝑟⋯𝑙_𝑛 or 𝑎₁₁𝑎₁₂⋯𝑎_1𝑟⋯𝑎_1𝑛𝑎₂₁𝑎₂₂⋯𝑎_2𝑟⋯𝑎_2𝑛⋯⋯⋯⋯⋯⋯𝑎_𝑛1𝑎_𝑛2⋯𝑎_𝑛𝑟⋯𝑎_𝑛𝑛 In the latter, or double suffix notation, the first suffix indicates the row, and the second the column. The former notation will be adopted in these pages.
A Composite determinant is one in which the number of columns exceeds the number of rows, and it is written as in the annexed example. 𝑎₁𝑎₂𝑎₃𝑏₁𝑏₂𝑏₃ Its value is the sum of all the determinants obtaied by taking a number of rows in every possible way.
A Simple determinant has single terms for its constituents.
A Compound determinant has more than one term in some or all of its constituents. See (570) for an example.
For the definitions of Symmetrical, Reciprocal, artial, and Complementary determinants; see (574), (575), and (576).

General Theory

556 The number of constituents is 𝑛². The number of elements in the complete determinant is |n. 557 The first or leading element is 𝑎₁𝑏₂𝑐₃⋯𝑙_𝑛. Any element may be derived from the first by permutation of the suffixes.
The sign of an element is + or − according as it has been obtained from the diagonal element by an even or odd number of permutations of the suffixes.
Hence the following rule for determining the sign of an element.

Rule

Take the suffixes in order, and put them back to their places in the first element. Let 𝑚 be the whole number of places passed over; then (−1)^𝑚 will give the sign required.

Example

To find the sign of the element 𝑎₄𝑏₃𝑐₅𝑑₁𝑒₂ of the determinant (𝑎₁𝑏₂𝑐₃𝑑₄𝑒₅).  𝑎4𝑏3𝑐5𝑑1𝑒2 Move the suffix 1, three places ⋯14352 Move the suffix 2, three places ⋯12435 Move the suffix 3, one place ⋯12345 In all, seven places; therefore (−1)⁷=−1 gives the sign required. 558 If two suffixes in any element be transposed, the sign of the element is changed.
Half of the elements are plus, and half are minus. 559 The elements are not altered by changing the rows into columns.
If two rows or columns are transposed, the sign of the determinant is changed. Because each element changes its sign.
If two rows or columns are identical, the determinant vanishes. 560 If all the constituents but one in a row or column vanish, the determinant becomes the product of that constituent and a determinant of the next lower order. 561 A cyclical interchange is effected by 𝑛−1 successive transpositions of adjacent rows or columns, until the top row has been brought to the bottom, or the left column to the right side. Hence
A cyclical interchange changes the sign of a determinant of an even order only.
The 𝑟^th row may be brought to the top by 𝑟−1 cyclical interchanges. 562 If each constituent in a row or column be multiplied by the same factor, the determinant becomes multiplied by it.
If each constituent of a row or column is the sum of 𝑚 terms, the compound determinant becomes the sum of 𝑚 simple determinants of the same order.
Also, if every constituent of the determinant consists of 𝑚 terms, the compound determinant is resolvable into the sum of 𝑚² simple determinants. 563 To express the minor of the 𝑟^th row and 𝑘^th column as a determinant of the 𝑛−1^th order.
Put all the constituents in the 𝑟^th row and 𝑘^th column equal to 0, and then make 𝑟−1 cyclical interchanges in the rows and 𝑘−1 in the columns, and multiply by (−1)^{(𝑟+𝑘)(𝑛−1)}. [∵ =(−1)^{(𝑟−1+𝑘−1)(𝑛−1)}. 564 To express a determinant as a determinant of a higher order.
continue the diagonal with constituents of "ones", and fill up with zeros on one side, and with any quantities whatever (𝛼, 𝛽, 𝛾, ⋯) on the other. 10000 𝛼1000 𝛽𝜖𝑎ℎ𝑔 𝛾𝜁ℎ𝑏𝑓 𝛿𝜂𝑔𝑓𝑐 565 The sum of the products of each constituent of a column by the corresponding minor in another given column is zero. And the same is true if we read 'row' instead of 'column'. Thus, referring to the determinant in (555), Taking the 𝑝^th and 𝑞^th columns, 𝑎_𝑝𝐴_𝑞+𝑏_𝑝𝐵_𝑞+⋯+𝑙_𝑝𝐿_𝑞=0 Taking the 𝑎 and 𝑐 rows, 𝑎₁𝐶₁+𝑎₂𝐶₂+⋯+𝑎_n𝐶_n=0 For in each case we have a determinant with two columns identical. 566 In any row or column the sum of the products of each constituent by its minor is the determinant itself. That is, Taking the 𝑝^th column, 𝑎_𝑝𝐴_𝑝+𝑏_𝑝𝐵_𝑝+⋯+𝑙_𝑝𝐿_𝑝=∆ Or taking the 𝑐 row, 𝑐₁𝐶₁+𝑐₂𝐶₂+⋯+𝑐_n𝐶_n=∆ 567 The last equation may be expressed by ∑𝑐_𝑝𝐶_𝑝=∆. Also, if (𝑎_𝑝𝐶_𝑞) express the determinant 𝑎_𝑝𝑎_𝑞𝑐_𝑝𝑐_𝑞; then ∑(𝑎_𝑝𝑐_𝑞) will represent the sum of all the determinants of the second order which can be formed by taking any two columns out of the 𝑎 and 𝑐 rows. The minor of (𝑎_𝑝, 𝑐_𝑞) may be written (𝐴_𝑝, 𝐶_𝑞), and signifies the determinant obtained by suppressing the two rows and two columns of 𝑎_𝑝 and 𝑐_𝑞. Thus ∆=∑(𝑎_𝑝, 𝑐_𝑞)(𝐴_𝑝, 𝐶_𝑞). And a similar notation when three or more rows and columns are selected. 568

Analysis of a Determinant

Rule

To resolve into its elements a determinant of the 𝑛^th order. Express it as the sum of 𝑛 determinants of the (𝑛−1)^th order by (560), and repeat the process with each of the new determinants.

Example

𝑎₁𝑎₂𝑎₃𝑎₄ 𝑏₁𝑏₂𝑏₃𝑏₄ 𝑐₁𝑐₂𝑐₃𝑐₄ 𝑑₁𝑑₂𝑑₃𝑑₄ =𝑎₁ 𝑏₂𝑏₃𝑏₄ 𝑐₂𝑐₃𝑐₄ 𝑑₂𝑑₃𝑑₄ −𝑎₂ 𝑏₃𝑏₄𝑏₁ 𝑐₃𝑐₄𝑐₁ 𝑑₃𝑑₄𝑑₁ +𝑎₃ 𝑏₄𝑏₁𝑏₂ 𝑐₄𝑐₁𝑐₂ 𝑑₄𝑑₁𝑑₂ −𝑎₄ 𝑏₁𝑏₂𝑏₃ 𝑐₁𝑐₂𝑐₃ 𝑑₁𝑑₂𝑑₃ Again, 𝑏₁𝑏₂𝑏₃ 𝑐₁𝑐₂𝑐₃ 𝑑₁𝑑₂𝑑₃ =𝑏₁ 𝑐₂𝑐₃ 𝑑₂𝑑₃ +𝑏₂ 𝑐₃𝑐₁ 𝑑₃𝑑₁ +𝑏₃ 𝑐₁𝑐₂ 𝑑₁𝑑₂ and so on. In the first series the derminants have alternately plus and minus signs, by the rule for cyclical interchanges (561), the order being even. 569

Synthesis of a determinant

The process is facilitated by making us of two evident rules. Those constituents which belong to the row and column of a given constituent 𝑎, will be designated "𝑎's constituents". Also , two pairs of constituents such as 𝑎_𝑝, 𝑐_𝑞 and 𝑎_𝑞, 𝑐_𝑝, forming the corners of a rectangle, will be said to be "conjugate: to each other. Rule I. No constituent will be found in the same term with one of its own constituents. Rule II> The conjugates of any two constituents 𝑎 and 𝑏 will be common to 𝑎's and 𝑏's constituents.

Example

To write the following terms in the form of a determinant: 𝑎𝑏𝑐𝑑+𝑏𝑓𝑔𝑙+𝑓²ℎ²+1𝑒𝑑𝑓+𝑐𝑔ℎ𝑝+1𝑎ℎ𝑟+𝑒𝑙𝑝𝑟−𝑓ℎ𝑝𝑟−𝑎𝑏𝑙𝑟−𝑎𝑐ℎ²−1𝑓ℎ𝑔−𝑏𝑑𝑓²−𝑒𝑓ℎ𝑙−𝑐𝑒𝑑𝑝 The determinant will be of the fourth order; and since every term must conatin four constituents, and the constituent 1 is supplied to make up the number in some of the terms. select any term, as 𝑎𝑏𝑐𝑑. for the leading diagonal.
Now apply Rule I., 𝑎 is not found with 𝑒, 𝑓, 𝑓, 𝑔, 𝑝, 0, ⋯1 𝑏 is not found with 𝑒, ℎ, ℎ, 𝑝, 1, 0, ⋯2 𝑐 is not found with 𝑓, 𝑓, 𝑙, 𝑟, 1, 0, ⋯3 𝑑 is not found with 𝑔, ℎ, ℎ, 𝑙, 𝑟, 0, ⋯4 Each constituent has 2(𝑛−1), that is, 6 constituents belonging to it, since 𝑛=4. Assuming, therefore, that the above letters are the constituents of 𝑎, 𝑏, 𝑐, and 𝑑, and that there are no more, we supply a sixth zero constituent in each case.
Now apply Rulle II. The constituents common to 𝑎 and 𝑏 are 𝑒, 𝑝; to 𝑎 and 𝑐 are 𝑓, 𝑓; to 𝑎 and 𝑑 are 𝑔, 0; to 𝑏 and 𝑐 are 1, 0 to 𝑏 and d are ℎ, ℎ, 0; to 𝑐 and d are 𝑙, 𝑟, 0; The determinant may now be formed. The diagonal being 𝑎𝑏𝑐𝑑; place 𝑒, 𝑝, the conjugates of 𝑎 and 𝑏, either as in the diagram or transposed.
Then 𝑓, and 𝑓, the conjugates of 𝑎 and 𝑐, may be written.
1 and 0, the conjugates of 𝑏 and 𝑐, must be placed as indicated, because 1 is one of 𝑝's constituents, since it is not found in any term with 𝑝, and must therefore be in the second row.
Similarly the places of 𝑔 and 0, and 𝑙, and 𝑟, are assigned.
In the case of 𝑏 and d we have ℎ, ℎ, 0 from which to choose the two conjugates, but we see that 0 is not one of them because that would assign two zero constituents to 𝑏, whereas 𝑏 has but one, which is already placed.
By similar reasoning the ambiguity in selecting the conjugates 𝑙, 𝑟 is removed.
𝑎𝑒𝑓𝑔 𝑝𝑏1ℎ 𝑓0𝑐𝑟 0ℎ𝑙d The foregoing method is rigid in the case of a complete determinant having different constituents. It becomes uncertain when the zero constituents increase in number, and when several constituents are identical. But even then, in the majority of cases, it will soon afford a clue to the required arrangement.

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive