Prove or Disprove the following a Let A be an n x n diagonal

Prove or Disprove the following:

a) Let A be an n x n diagonalizable matrix such that all its eigenvalues are equal to a scalar c.

Then necessarily A = c I.

b) Let A be an n x n matrix such that all its eigenvalues are equal to a scalar c.

Then necessarily A = c I.

c) Let A be an n x n diagonalizable matrix such that A^k = 0 for some positive integer k. Prove that A = 0.

d) Construct an example of a 2 x 2 matrix A that has no zero entries but is such that A^k = 0 for some positive integer k.

Solution

1) let A= 3x3 matrix

it is diagnolizable only if its algebric multiplicity equals to the goemetric multiplicity

given that its all eigen value equal to scalar c and equals .

Hence its algebric multiplicity = 2 ( two times reapeted )

gomentric multiplicity equal to number of independent solution .

since its order is 3 hence its rank should be 1 so that number of independent solution = 3-1 =2 which has to be equal to AM. as matrix is diagnilizable

for rank 1 and all eigen value equal to same scalar, matrix must be a scalar matrix

Hence A= cI

2) no there is no such compulsition on matrix as its diagnozibility is not defined

3) Ak, where k is some positive integer

if Ak=0 its posible only when A=0

d) thats not posible because if multiplication result is zero then either of one operands has to be zero

since k is some positive integer hence A=0