5 In each question circle either True of False No justificat

5.

In each question circle either True of False. No justification Ls needed. Let A and B be m Times n matrices. If A is row-equivalent to B. then Col(A) = Col(B). Distinct eigenvectors are linearly independent. If 0 is an eigenvalue of an n Times n matrix A. then rank(A)

Solution

5)

a) True.

Since both are mxn, if rows are equal column have to be equal

b) True. Distinct eigen vectors are always linearly independent.

c) True. If 0 is eigen value then matrix A is singular. Hence by row operation atleast one row can be made 0.

This implies rank of A <n

d) True. Complex roots cannot be equal and hence distinct eigen vectors. So diagonalisable.

e) False. If an eigen value with multiplicity of more than 1, has distinct eigen vectors then the matrix can be diagnolised.

f) True