The determinant `|(b_1+c_1,c_1+a_1,a_1+b_1),(b_2+c_2,c_2+a_2,a_2+b_2),(b_3+c_3,c_3+a_3,a_3+b_3...

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The determinant |[ b_1+c_1 c_1+a_1 a_1+b_1; b_2+c_2 c_2+a_2 a_2+b_2; b_3+c_3 c_3+a_3 a_3+b_3 ]|=A... The determinant |[ b_1+c_1 c_1+a_1 a_1+b_1; b_2+c_2 c_2+a_2 a_2+b_2; b_3+c_3 c_3+a_3 a_3+b_3 ]|=A...
if `D_1=|{:(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3):}|` and `D_2=|{:(a_1+2a_2+3a_3,2a_3,5a_2) if `D_1=|{:(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3):}|` and `D_2=|{:(a_1+2a_2+3a_3,2a_3,5a_2)
Let `D= |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|, D_1=|(a_1+pb_1, b_1+qc_1, c1+ra_1) Let `D= |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|, D_1=|(a_1+pb_1, b_1+qc_1, c1+ra_1)
IfDelta=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)| and A_2,B_2,C_2 are respectively cofactors o... IfDelta=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)| and A_2,B_2,C_2 are respectively cofactors o...
If triangle=|[a_1+b_1x,a_1x+b_1,c_1],[a_2+b_2x,a_2x+b_2,c_2],[a_3+b_3x,a_3x+b_3,c_3]|=0, then: |... If triangle=|[a_1+b_1x,a_1x+b_1,c_1],[a_2+b_2x,a_2x+b_2,c_2],[a_3+b_3x,a_3x+b_3,c_3]|=0, then: |...
If `Delta=|[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]|` and `A_1,B_1,C_1` denote the co-factors If `Delta=|[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]|` and `A_1,B_1,C_1` denote the co-factors
If Δ=|[ a_1 b_1 c_1; a_2 b_2 c_2; a_3 b_3 c_3 ]| and A_2, B_2, C_2 are respectively cofactors of ... If Δ=|[ a_1 b_1 c_1; a_2 b_2 c_2; a_3 b_3 c_3 ]| and A_2, B_2, C_2 are respectively cofactors of ...
Confirm the identities without evaluating the determinants directly. |[ a_1… ] Confirm the identities without evaluating the determinants directly. |[ a_1… ]