If `A=[a_("ij")]_(n xx n)` is such that `a_("ij")= bar(a)_("ji") AA I, j` and `A^(2)=O`, then prove that matrix A is null matrix. Here, `bar(a)_("ji")` denotes the conjugate `a_("ji")`.

Statement 1: If A=([a_(i j)])_(nxxn) is such that ( a )_(i j)=a_(j i),AAi ,ja n dA^2=O , then matrix A null matrix. Statement 2: |A|=0.

A=[a_(ij)]_(mxxn) is a square matrix if …………

If a square matix a of order three is defined A=[a_("ij")] where a_("ij")=s g n(i-j) , then prove that A is skew-symmetric matrix.

If A=(a_(ij))_(2xx2) .Where a_(ij) is given by (i-2j)^(2) then A is:

If A=[a_("ij")]_(4xx4) , such that a_("ij")={(2",","when "i=j),(0",","when "i ne j):} , then {("det (adj (adj A))")/(7)} is (where {*} represents fractional part function)

If A=[a_(ij)]_(mxxn) and a_(ij)=(i^(2)+j^(2)-ij)(j-i) , n odd, then which of the following is not the value of Tr(A)

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

Let A=[a_("ij")] be a matrix of order 2 where a_("ij") in {-1, 0, 1} and adj. A=-A . If det. (A)=-1 , then the number of such matrices is ______ .

If matrix A=[a_(ij)]_(3xx) , matrix B=[b_(ij)]_(3xx3) , where a_(ij)+a_(ji)=0 and b_(ij)-b_(ji)=0 AA i , j , then A^(4)*B^(3) is