Page:Elementary algebra (1896).djvu/454

that is, if we interchange two rows or two columns of the determinant, we obtain a determinant which differs from it only in sign.

544. Let us now consider the homogeneous linear equations

a_1x + b_1y + c_1 z = 0, a_2x + b_2y + c_2 z = 0, a_3x + b_3y + c_3 z = 0,

By eliminating x, y, z, we obtain

or

This is usually written

and the expression on the left being a determinant which

consists of three rows and three columns is called a determinant of the third order.

545. By a rearrangement of terms, the expanded form of the above determinant may be written

or hence

that is, the value of the determinant is not altered by changing the rows into columns, and the columns into rows.