have evidently the solution 0, 0. Let us inquire if they can be simultaneously satisfied by values of x and y other than 0, 0. Multiplying (1) by b2, and (2) by dg,

if

This equation will be satisfied by a value of x other than 0,

The same result would have been obtained, had we first eliminated y.

is an example of a form which occurs frequently in mathematics, and for which a special notation has been devised. Such an expression is called a Determinant.

The determinant (I.) is usually written in a square form:

b1 ba

The positive term of the determinant, obtained from the square form, is the cross-product from upper left to lower right; the negative term is the cross-product from upper right to lower left.

3. We shall frequently call the symbolic form in which the determinant is written the determinant.

The advantage of writing determinants in this form will be made evident in subsequent work.

4. The Elements of a determinant are the unconnected symbols in its square form; as a1, ɑ2, b1, b2 in (I.).

A Row of a determinant is a horizontal line of elements; as a1, a2.

A Column of a determinant is a vertical line of elements; as

The rows are numbered first, second, etc., counting from top to bottom; and the columns from left to right. The square form has two diagonals.

The Principal Diagonal is composed of the elements from the

upper left-hand corner to the lower right-hand corner; as

in (I.).

ar

b2

The Order of a determinant is the number of rows or columns in its square form.

are satisfied by values of x, y, z, other than 0, 0, 0. Multiplying (2) by c„, and (3) by b

Subtracting, (b1C3 − b2€1)x + (b¿C, — b¿C2)y = 0.

(1)

(2)

(3)

In like manner, from (2) and (3), we obtain

Substituting these values of y and z in (1),

or,

[a1 (b.c3b3c2) — а2(b1C3 — bgC1) + аg (b12 — b ̧¤1)] x = 0.

This equation will be satisfied by a value of x other than 0, if the expression

The expression (II.) is called a determinant of the third order, and is usually written in the square form

Minors.

6. If the row and column in which a, stands, in the determinant (IV.), be deleted, the remaining elements constitute a determinant of the second order, ხიხვი C2 C3

This determinant is called the Minor of the given determinant with respect to a1, and may be denoted by A1.

If the row and column in which a2 stands be deleted, the b1 bg remaining elements constitute the determinant

This determinant, with sign changed, is called the minor with respect to a, and is denoted by Ag.

In like manner the minor with respect to a is

and is

denoted by Ag.

7. The form of the determinant of the third order given in (III.), Art. 5, can now be written

Written in this form, the determinant is said to be expanded in terms of the elements of the first row and the corresponding minors. By this expansion the value of any determinant of the third order can be readily obtained.

2(28 + 12)+3(35 − 2) + (30 + 4)=213.

8. The determinant of the third order can also be expanded in terms of the elements of any row, or column, and the corresponding minors.

For, rearranging the terms of the determinant (II.), Art. 5, we have

In (V.) and (VI.), — (α¿Cз — α ̧¤2) is the minor with respect to

b, and is denoted by B1, etc.

The above expansions can now be written

In general, the minor with respect to any element is, except for sign, the determinant obtained by deleting the row and column in which the element stands.

The signs to be prefixed to the determinants thus obtained alternate + and -, beginning with a, and going either horizontally or vertically, but not diagonally. Thus, the sign of the minor of a1 is +, of b1 is of b2 is, of b, is, of c, is

Since the second row contains a zero element, the work is simplified by expanding in terms of the elements of this row:

9. The terms of a determinant of the third order can also be obtained directly from the square form. Removing parentheses in (II.), Art. 5, and writing positive terms first, we have

abcg + იხვს + ვba — abc — სხვ — ვხას.

The terms are obtained by multiplying together the elements connected by continuous lines in the following schemes: