Let `p=[(3,-1,-2),(2,0,alpha),(3,-5,0)],` where `alpha in RR.` Suppose `Q=[q_(ij)]` is a matrix such that `PQ=kI,` where `k in RR, k != 0 and I`is the identity matrix of order 3. If `q_23=-k/8 and det(Q)=k^2/2,` then

Let A=[("tan"pi/3,"sec" (2pi)/3),(cot (2013 pi/3),cos (2012 pi))] and P be a 2 xx 2 matrix such that P P^(T)=I , where I is an identity matrix of order 2. If Q=PAP^(T) and R=[r_("ij")]_(2xx2)=P^(T) Q^(8) P , then find r_(11) .

If M is a 3 xx 3 matrix, where det M=1 and MM^T=1, where I is an identity matrix, prove theat det (M-I)=0.

Let A be a square matrix of order 2 such that A^(2)-4A+4I=0 , where I is an identity matrix of order 2. If B=A^(5)+4A^(4)+6A^(3)+4A^(2)+A-162I , then det(B) is equal to _________

If A is square matrix of order 3 matrix, |A| != 0 and |3A|=k|A| , then write the value of k .

If A=[(1, 0,-3 ),(2, 1 ,3 ),(0, 1 ,1)] , then verify that A^2+A=A(A+I) , where I is the identity matrix.

Let A=[(0, i),(i, 0)] , where i^(2)=-1 . Let I denotes the identity matrix of order 2, then I+A+A^(2)+A^(3)+……..A^(110) is equal to

A square matrix [a_(ij)] such that a_(ij)=0 for i ne j and a_(ij) = k where k is a constant for i = j is called _____

Let P=[[1,0,0],[4,1,0],[16,4,1]] and I be the identity matrix of order 3 . If Q = [q_()ij ] is a matrix, such that P^(50)-Q=I , then (q_(31)+q_(32))/q_(21) equals

If P=[(6,-2),(4,-6):}] and Q = [{:(5,3),(2,0):}] find the matrix M such that 2Q - 3P - 3M =0

If A=(3 5 7 9) is written as A = P + Q, where P is a symmetric matrix and Q is skew symmetric matrix, then write the matrix P.