We consider two equations:


 a1x+ b1y = 0

and a2x+ b2y = 0,

From these two equations, we have

Now, we shall express the above eliminant as

Here, we suppressed the letters x and y to be eliminated and enclosed their coefficients as above in two parallel lines. The left hand member of (i) is called determinant of second order and its value is

 a1b2+ a2b1 = 0
Similarly, we consider three equations

 a1x+ b1y + c1z = 0

a2x+ b2y + c2z = 0

a3x+ b3y + c3z = 0

On eliminating x, y, z from the above equations we shall have

a1 (b2c3 − b3c2) −  b1(a2 + c3 − a3c2)  +  c1(a2b3 − b2a3)   .. (ii)

which can be expressed in form of second order determinant as

The above expression is the expanded form of the following

The above expression is called a third order determinant expression (ii) is the expansion of the determinant (iii) along the first row.