Show that if `x_1, x_2, x_3!=0` `|x_1+a_1b1a_1b_2a_1b_3a_2b_1x_2+a_2b_2a_2b_3a_3b_1a_3b_2x_3+a_3b_3|=x_1x_2x_3(1+(a_1b_1)/(x_1)+(a_2b_2)/(x_2)+(a_3b_3)/(x_3))` .

Show that if x_(1),x_(2),x_(3) ne 0 |{:(x_(1) +a_(1)b_(1),,a_(1)b_(2),,a_(1)b_(3)),(a_(2)b_(1),,x_(2)+a_(2)b_(2),,a_(2)b_(3)),(a_(3)b_(1),,a_(3)b_(2),,x_(3)+a_(3)b_(3)):}| =x_(1)x_(2)x_(3) (1+(a_(1)b_(1))/(x_(1))+(a_(2)b_(2))/(x_(2))+(a_(3)b_(3))/(x_(3)))

If a_i, b_i in N for i 1,2,3, then coefficient of x in the determinant; |((1+x)^(a_1b_1),(1+x)^(a_1b_2),(1+x)^(a_1b_3)),((1+x)^(a_2b_1),(1+x)^(a_2b_2),(1+x)^(a_2b_3)), ((1+x)^(a_3b_1),(1+x)^(a_3b_2),(1+x)^(a_3b_3))|

If |x_1y_1 1x_2y_2 1x_3y_3 1|=|a_1b_1 1a_2b_2 1a_3b_3 1| then the two triangles with vertices (x_1, y_1),(x_2,y_2),(x_3,y_3) and (a_1,b_1),(a_2,b_2),(a_3,b_3) are equal to area (b) similar congruent (d) none of these

If |x_1y_1 1x_2y_2 1x_3y_3 1|=|a_1b_1 1a_2b_2 1a_3b_3 1| then the two triangles with vertices (x_1, y_1),(x_2,y_2),(x_3,y_3) and (a_1,b_1),(a_2,b_2),(a_3,b_3) are equal to area (b) similar congruent (d) none of these

The value of |(a_(1) x_(1) + b_(1) y_(1),a_(1) x_(2) + b_(1) y_(2),a_(1) x_(3) + b_(1) y_(3)),(a_(2) x_(1) +b_(2) y_(1),a_(2) x_(2) + b_(2) y_(2),a_(2) x_(3) + b_(2) y_(3)),(a_(3) x_(1) + b_(3) y_(1),a_(3) x_(2) + b_(3) y_(2),a_(3) x_(3) + b_(3) y_(3))| , is

If a x_1^2+b y_1^2+c z_1^2=a x_2 ^2+b y_2 ^2+c z_2 ^2=a x_3 ^2+b y_3 ^2+c z_3 ^2=d ,a x_2 x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1=a x_1x_2+b y_1y_2+c z_1z_2=f, then prove that |(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3,y_3,z_3)|=(d-f){((d+2f))/(a b c)}^(1//2)

Find the coefficient of x in the determinant |{:((1+x)^(a_(1)b_(1)),(1+x)^(a_(1)b_(2)),(1+x)^(a_(1)b_(3))),((1+x)^(a_(2)b_(1)),(1+x)^(a_(2)b_(2)),(1+x)^(a_(2)b_(3))),((1+x)^(a_(3)b_(1)),(1+x)^(a_(3)b_(2)),(1+x)^(a_(3)b_(3))):}|

If a x1 2+b y1 2+c z1 2=a x1 2+b y2 2+c z2 2=a x3 2+b y3 2+c z3 2=d\ a n d\ a x_2x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1=a x_1x_2+b y_1y_2+c z_1z_2=f then prove that |x_1y_1z_1x_2y_2z_1x_3y_3z_3|=(d-f)[(d+2f)/(a b c)]^(//2)(a , b ,\ c!=0)

Let phi_1(x)=x+a_2, phi_2(x)=x^2+b_1x+b_2, x_1=2,x_2=3 and x_3=5 and Delta=|(1,1,1),( phi_1(x_1), phi_1(x_2), phi_1(x_3)),(phi_2(x_1),phi_2(x_2),phi_2(x_3))| . Find the value of Delta .

If a_(1) x^(3) + b_(1)x^(2) + c_(1)x + d_(1) = 0 and a_(2)x^(3) + b_(2)x^(2) + c_(2)x + d_(2) = 0 a pair of repeated roots common, then prove that |{:(3a_(1)", "2b_(1) ", "c_(1)),(3a_(2)", " 2b_(2)", "c_(2)),(a_(2)""b_(1)- a_(1)b_(2)", "c_(2)a_(1)-c_(2)a_(1)", "d_(1)a_(2)-d_(2)a_(1)):}|=0