Determine whether each of the following statements is TRUE o

Determine whether each of the following statements is TRUE or FALSE. If every eigenvalue of a matrix A has algebraic multiplicity 1, then A is diagonalizable. Singular matrices are not diagonalizable. If a matrix A is diagonalizable, then so is A^k for every positive integer k.

Solution

a) if algebraic multiplicity of eigenvalue is 1 then algebraic multiplicity is equal to geometrix multiplicity

hence matrix is diagonalizable

statement is true

b) a matrix is singular if its one eigenvalue is 0

hence a singular matrix can be diagonalizable or it cannot be diagonalizable

the statement is false

c) if a matrix A is diagonalizable then A^k is also diagonalizable for any positive integer k

statement is true