Find the value of `lambda` for which `|{:(2a_1+b_1 , 2a_2+b_2 , 2a_3+b_3),(2b_1+c_1, 2b_2+c_2 , 2b_3+c_3),(2c_1+a_1,2c_2+a_2, 2c_3+a_3):}|=lambda|{:(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3):}|`

|{:(a,a^2,bc),(b,b^2,ca),(c,c^2,ab):}|=|{:(1,a^2,a^3),(1,b^2,b^3),(1,c^2,c^3):}|

Let Delta_1=|{:(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3):}| , Delta_2=|{:(6a_1,2a_2,2a_3),(3b_1,b_2,b_3),(12c_1, 4c_2,4c_3):}| and Delta_3=|{:(3a_1+b_1 , 3a_2+b_2 , 3a_3+b_3),(3b_1,3b_2,3b_3),(3c_1,3c_2,3c_3):}| then Delta_3-Delta_2=kDelta_1 , find k.

Without expanding the determinant, prove that {:|( a, a ^(2), bc ),( b ,b ^(2) , ca),( c, c ^(2) , ab ) |:} ={:|( 1, a^(2) , a^(3) ),( 1,b^(2) , b^(3) ),( 1, c^(2),c^(3)) |:}

If Delta=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)| and Delta_1=|(a_1+pb_1,b_1+qc_1,c_1+ra_1),(a_2+pb_2,b_2+qc_2,c _2+ra_2),(a_3+pb_3,b_3+qc_3,c_3+ra_3)| then Delta_1=

Find the coordinates of the centriod of the triangle whose vertices are ( a_(1), b_(1), c_(1)) , (a_(2), b_(2), c_(2)) and (a_(3), b_(3), c_(3)) .

If the points ( a_1,b_1),(a_2,b_2) and (a_1+a_2,b_1+b_2) are collinear ,show that a_1b_2=a_2b_1 .

Prove that the value of each the following determinants is zero: |[a_1,l a_1+mb_1,b_1],[a_2,l a_2+mb_2,b_2],[a_3,l a_3+m b_3,b_3]|

If x ,y \ a n d \ z are not all zero and connected by the equations a_1x+b_1y+c_1z=0,a_2x+b_2y+c_2z=0 , and (p_1+lambdaq_1)x+(p_2+lambdaq_2)+(p_3+lambdaq_3)z=0 , show that lambda=-|[a_1,b_1,c_1],[a_2,b_2,c_2],[p_1,p_2,p_3]|-:|[a_1,b_1,c_1],[a_2,b_2,c_2],[q_1,q_2,q_3]|

Prove that, |(2a, a-b-c,2a),(2b,2b,b-c-a),(c-a-b,2c,2c)|=(a + b+c)^3

prove that, |{:(" "2a,a-b-c," "2a),(" "2b," "2b,b-c-a),(c-a-b," "2c," "2c):}|=(a+b+c)^3