Basic Properties of Determinants

9. Determinant of Cofactor Matrix

Δ =

∣∣∣∣x11x21x31x12x22x32x13x23x33∣∣∣∣|x11x12x13x21x22x23x31x32x33|

then Δ

11

=

∣∣∣∣z11z21z31z12z22z32z13z23z33∣∣∣∣|z11z12z13z21z22z23z31z32z33|

= Δ

22

In the above determinants of the cofactor matrix,C_ij denotes the cofactor of the elements a_ij in Δ.

10. Property of Invariance

∣∣∣∣a1a2a3b1b2b3c1c2c3∣∣∣∣|a1b1c1a2b2c2a3b3c3|

=

∣∣∣∣a1+αb1+βc1a2+αb2+βc2a3+αb3+βc3b1b2b3c1c2c3∣∣∣∣|a1+αb1+βc1b1c1a2+αb2+βc2b2c2a3+αb3+βc3b3c3|

 

It implies that determinant remains unchanged under an operation of the term C_i ⟶ C_i + αC_j + βC_kj where, j and k is not equivalent to i, or a Mathematical operation of the term R_i ⟶ R_i + αR_j  + βR_k, where, j and k is not equivalent to i.