Without expanding the determinants,show that`|(a,a+b,a+2b),(a+2b,a,a+b),(a+b,a+2b,a)|=9(a+b)b^2`

Without expanding the determinants,show that |(b+c,a-b,a),(c+a,b-c,b),(a+b,c-a,c)|=3abc-a^3-b^3-c^3

Without expanding determinant show that |(0,a,b),(-a,0,c),(-b,-c,0)|=0

Without expanding determinant show that |(b+c,bc,b^2c^2),(c+a,ca,c^2a^2),(a+b,ab,a^2b^2\)|=0

Without expanding determinant show that |(xa,yb,zc),(a^2,b^2,c^2),(1,1,1)|=|(x,y,z),(a,b,c),(bc,ca,ab)|

Using properties of determinant show that |(a+b,a,b),(a,a+c,c),(b,c,b+c)|=4abc

Without expanding evaluate the following determinants. |(1,a,b+c),(1,b,c+a),(1,c,a+b)|

Without expanding,show that the values of the following determinants are zero: |(a,b,c),(2,5,7),(a+4,b+10,c+14)|