Page:Elementary algebra (1896).djvu/453

 CHAPTER XLVII.

Determinants.

542. Consider two homogeneous linear equations

multiplying the first equation by $$b_2$$, the second by $$b_1$$, subtracting and dividing by $$x$$, we obtain

This result is sometimes written

and the expression on the left is called a determinant. It consists of two rows and two columns, and in its expanded form or development, as seen in the first member of (1), each term is the product of two quantities; it is therefore said to be of the second order. The line $$a_1b_2$$ is called the principal diagonal, and the line $$b_1a_2$$, the secondary diagonal.

The letters $$a_1, b_1, a_2, b_2$$, are called the constituents of the determinant, and the terms $$a_1b_2, a_2b_1$$ are called the elements.

THE VALUE OF THE DETERMINANT AFTER CERTAIN CHANGES.

543. Since $$\begin{vmatrix}a_1 & b_1 \\ a_2 & b_2\end{vmatrix} = a_1b_2 - a_2b_1 = \begin{vmatrix}a_1 & a_2 \\ b_1 & b_2\end{vmatrix}$$,

it follows that the value of the determinant is not altered by changing the rows into columns, and the columns into rows. Again, it is easily seen that $$\begin{vmatrix}a_1 & b_1 \\ a_2 & b_2\end{vmatrix} = -\begin{vmatrix}b_1 & a_1 \\ b_2 & a_2\end{vmatrix}$$, and $$\begin{vmatrix}a_1 & b_1 \\ a_2 & b_2\end{vmatrix} = -\begin{vmatrix}a_2 & b_2 \\ a_1 & b_1\end{vmatrix}$$;