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We consider two equations:

a_{1}x+ b_{1}y_{ }= 0

and a_{2}x+ b_{2}y_{ }= 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

a_{1}b_{2}+ a_{2}b_{1}_{ }= 0

Similarly, we consider three equations

a_{1}x+ b_{1}y + c_{1}z_{ }= 0

a_{2}x+ b_{2}y + c_{2}z_{ }= 0

a_{3}x+ b_{3}y + c_{3}z_{ }= 0

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

a_{1} (b_{2}c_{3 − }b_{3}c_{2}) − b_{1}(a_{2 }+_{ }c_{3 − }a_{3}c_{2}) + c_{1}(a_{2}b_{3 − }b_{2}a_{3}) .. (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.

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