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Let =|2a1b1a1b2+a2b1a1b3+a3b1a1b2+a2b1 2...

Let `=|2a_1b_1a_1b_2+a_2b_1a_1b_3+a_3b_1a_1b_2+a_2b_1 2a_2b_2a_2b_3+a_3b_2a_1b_3+a_3b_1a_3b_2+a_2b_3 2a_3b_3|` . Expressing `` as the product of two determinants, show that `=0.` Hence, show that if `a x^2+2h x y+b y^2+2gx+2fy+c=(l x+m y+n)(l^(prime)x+m^(prime)y+n),t h e n|a hgh bfgfc|=0.`

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Let triangle =|[2a_1b_1, a_1b_2+a_2b_1, a_1b_3+a_3b_1], [a_1b_2+a_2b_1, 2a_2b_2, a_2b_3+a_3b_2], [a_1b_3+a_3b_1, a_3b_2+a_2b_3, 2a_3b_3]| . Expressing as the product of two determinants, show that triangle=0

Let triangle =|[2a_1b_1, a_1b_2+a_2b_1, a_1b_3+a_3b_1], [a_1b_2+a_2b_1, 2a_2b_2, a_2b_3+a_3b_2], [a_1b_3+a_3b_1, a_3b_2+a_2b_3, 2a_3b_3]| . Expressing as the product of two determinants, show that triangle=0

|[2a_1b_1, a_1b_2+a_2b_1, a_1b_3+a_3b_1] , [a_1b_2+a_2b_1, 2a_2b_2, a_2b_3+a_3b_2] , [a_1b_3+a_3b_1, a_3b_2+a_2b_3, 2a_3b_3]|=

Let = |(2a_(1)b_(1),a_(1)b_(2)+a_(2)b_(1),a_(1)b_(3)+a_(3)b_(1)),(a_(1)b_(2)+a_(2)b_(1),2a_(2)b_(2),a_(2)b_(3)+a_(3)b_(2)),(a_(1)b_(3)+a_(3)b_(1),a_(3)b_(2)+a_(2)b_(3),2a_(3)b_(3))| Express the determinant D as a product of two determinants. Hence or otherwise show that D = 0.

Show that | (a_1l_1+b_1m_1, a_1l_2+b_1m_2,a_1l_3+b_1m_3),(a_2l_1+b_2m_1, a_2l_2+b_2m_2, a_2l_3+b_2m_3),(a_3l_1+b_3m_1, a_3l_2+b_3m_2,a_3l_3+b_3m_3)| = 0

Prove that : {:|(a_1x_1+b_1y_1,a_1x_2+b_1y_2,a_1x_3+b_1y_3),(a_2x_1+b_2y_1,a_2x_2+b_2y_2,a_2x_3+b_2y_3),(a_3x_1+b_3y_1,a_3x_2+b_3y_2,a_3x_3+b_3y_3)|

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!=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)) .