NET EXAM JUNE 2007

Important question for mathematics ( NET) 2007
MARK TRUE OR FALSE. Here [ ] represent coloum vector.
1.if A is a invertible matrix, then A*(-1 ) i.e.inverse of and A*(T) i.e transpose of A have same eigen values.
2.all symmetric real matrices are diagonalizable.
3.there exist a symmetric 2*2 matrix A ,such that A[1,2]=[3,6] and A[1,1]=[2,2].
4.Find the matrix A ,such that [1,2] is the eigen vector with eigen value 2 and [1,3] is the eigen vector of A with eigen value 3.
5.Euler line passes through through centriod, circumcentre and orthocenter of the triangle.
6.if a real 2*2 matrix has i has an eigenvalue, then it is orthogonal.
7.if A=BCD, where A,B,C,D are 3 by 3 not to be equal matrices and A is not invertiable, then one of them is not invertiable.
8.if A is a symmetric matrix, such that A*5( power 5)=0, then A=0.
9.if A is a non-zero diagonlizable 4 by 4 matrices , then A*4(power 4) is non zero, then A is non-zero matrix is nilpotent iff all its eigenvalues to zero.
10.if two matrices are symmetric and same eigenvalues (with algebraic multiplicities) , then they are similar.