Page:Elementary algebra (1896).djvu/453

CHAPTER XLVII.

Determinants.

542. Consider two homogeneous linear equations

$a_{1}x+b_{1}y=0,$

$a_{2}x+b_{2}y=0$;

multiplying the first equation by $b_{2}$, the second by $b_{1}$, subtracting and dividing by $x$, we obtain

$a_{1}b_{2}-a_{2}b_{1}=0\ldots$ (1)

This result is sometimes written

${\begin{vmatrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{vmatrix}}=0$,

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 $a_{1}b_{2}$ is called the principal diagonal, and the line $b_{1}a_{2}$, the secondary diagonal.

The letters $a_{1},b_{1},a_{2},b_{2}$, are called the constituents of the determinant, and the terms $a_{1}b_{2},a_{2}b_{1}$ are called the elements.

THE VALUE OF THE DETERMINANT AFTER CERTAIN CHANGES.

543. Since ${\begin{vmatrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{vmatrix}}=a_{1}b_{2}-a_{2}b_{1}={\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}$,

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 ${\begin{vmatrix}a_{1}&b_{1}\\a_{2}&b_{2}\end{vmatrix}}=-{\begin{vmatrix}a_{2}&b_{2}\\a_{1}&b_{1}\end{vmatrix}}$;

435