Basic Properties of Determinants

6. Scalar Multiple Property

If all the elements of a row (or columns) of a determinant is multiplied by a non-zero constant, then the determinant gets multiplied by a similar constant.

7. Sum Property

∣∣∣∣j1+k1j2+k2j3+k3l1l2l3m1m2m3∣∣∣∣|j1+k1l1m1j2+k2l2m2j3+k3l3m3|

=

∣∣∣∣j1j2j3l1l2l3m1m2m3∣∣∣∣|j1l1m1j2l2m2j3l3m3|

+

∣∣∣∣+k1+k2+k3l1l2l3m1m2m3∣∣∣∣|+k1l1m1+k2l2m2+k3l3m3|

8. Triangle Property

If each term of a determinant above or below the main diagonal comprise zeroes, then the determinant is equivalent to the product of diagonal terms. That is

∣∣∣∣x100x2y20x3y3z3∣∣∣∣|x1x2x30y2y300z3|

=

∣∣∣∣x1x2x30y2y300z3∣∣∣∣|x100x2y20x3y3z3|

= X

11

Y

22

Z

33