Add the following: `a-b+ab` ,`b-c+bc` ,c-a+ac`

Add the following: ab-bc-ca , ca-ab

Using the Properties of determinants, prove the following: {:|(1,1,1),(a,b,c),(bc,ca,ab)|=(a-b)(b-c)(c-a)

If (lna)/(b-c)=(lnb)/(c-a)=(lnc)/(a-b) , prove the following . a^(b^2+bc+c^2).b^(c^2+ca+a^2).c^(a^2+ab+b^2)=1

Using the Properties of determinants, prove that following: {:|(-a^2,ab,ac),(ba,-b^2,bc),(ac,bc,-c^2)|=4a^2b^2c^2

If (Ina)/(b-c)=(Inb)/(c-a)=(Inc)/(a-b) , prove the following . a^a.b^b.c^c=1

Add: ab-4a,4b-ab,4a-4b

Using the property of determinants prove that {:|( 3a,-a+b,-a+c),( -b+a, 3b,-b+c) ,( -c+a,-c+b,3c) |:} = 3( a+b+c) ( ab+bc+ca)

Find the proudct (a-b-c) (a ^(2) + b ^(2) + c ^(2) - ab + bc-ca) without actual multiplication.

Using the property of determinants and without expanding prove that {:|( -a^(2) , ab,ac),( ba,-b^(2) , bc) ,( ca, cb, -c^(2)) |:} =4a^(2) b^(2) c^(2)

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)) |:}