If A=`|(3,2,-1),(1,0,2),(-2,1,2)|`Prove that
(i)`a_(11)A_(11)+a_(12)A_(12)+a_(13)A_(13)=det(A)`
(ii)`a_(21)A_(11)+a_(22)A_(12)+a_(23)A_(13)=0`

If A = [{:(3," "1,-1),(2,-2," "0),(1," "2,-1):}] "and"" "A^(-1)=[(a_(11),a_(12),a_(13)),(a_(21),a_(22),a_(23)),(a_(31),a_(32),a_(33))] then the value of a_(23) is

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)))

Let A=([a_(i j)])_(3xx3) be a matrix such that AA^T=4Ia n da_(i j)+2c_(i j)=0,w h e r ec_(i j) is the cofactor of a_(i j)a n dI is the unit matrix of order 3. |a_(11)+4a_(12)a_(13)a_(21)a_(22)+4a_(23)a_(31)a_(32)a_(33)+4|+5lambda|a_(11)+1a_(12)a_(13)a_(21)a_(22)+1a_(23)a_(31)a_(32)a_(33)+1|=0 then the value of 10lambda is _______.

If the sequence a_(1),a_(2),a_(3),…,a_(n) is an A.P., then prove that a_(1)^(2)-a_(2)^(2)+a_(3)^(2)-a_(4)^(2)+…+a_(2n-1)^(2)-a_(2n)^(2)=n/(2n-1)(a_(1)^(2)-a_(2n)^(2))

If A+iB=(a_(1)+ib_(1))(a_(2)+ib_(2))(a_(3)+ib_(3)) then A^(2)+B^(2) is :

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_(1)),(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

Let A=[a_("ij")]_(3xx3) be a matrix such that A A^(T)=4I and a_("ij")+2c_("ij")=0 , where C_("ij") is the cofactor of a_("ij") and I is the unit matrix of order 3. |(a_(11)+4,a_(12),a_(13)),(a_(21),a_(22)+4,a_(23)),(a_(31),a_(32),a_(33)+4)|+5 lambda|(a_(11)+1,a_(12),a_(13)),(a_(21),a_(22)+1,a_(23)),(a_(31),a_(32),a_(33)+1)|=0 then the value of lambda is

if a_(1),a_(2),a+_(3)……,a_(12) are in A.P and Delta_(1)= |{:(a_(1)a_(5),,a_(1),,a_(2)),(a_(2)a_(6),,a_(2),,a_(3)),(a_(3)a_(7),,a_(3),,a_(4)):}| Delta_(3)= |{:(a_(2)b_(10),,a_(2),,a_(3)),(a_(3)a_(11),,a_(3),,a_(4)),(a_(3)a_(12),,a_(4),,a_(5)):}| then Delta_(2):Delta_(2)= "_____"

If A_(i),B_(i),C_(i) are the cofactors of a_(i),b_(i),c_(i) respectively,i=1,2,3 in Delta =|(a_(1),b_(2),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| show that |(A_(1),B_(1),C_(1)),(A_(2),B_(2),C_(2)),(A_(3),B_(3),C_(3))|=Delta^(0)