Question

1) True or False

a. The sum of the dimensions of the eigenspaces of the distinct egenvalues of a matrix of order n can sometimes be bigger than n.

b. A square matrix A is singular if and only if 0 is an eigenvalue of A.

c. If two eigenvectors of a matrix are linearly independent, then they correspond to distinct eigenvalues.

d. Two square matrices are similar if and only if they have the same eigenvalues and the same rank.

e. A square matrix A is diagonalizable if and only if for each eigenvalue c of A, the algebraic multiplicity of c is equal to the dimension of the eigenspace of A corresponding to c.

Answer 1

Solution:

(a)

The answer will be "False"

Explanation:

=>The sum of the dimensions of the eigenspaces of the distrinct eigenvalue of matrix of order is always less than n.

(b)

The answer will be "True"

Explanation:

=>If matrix A is singular then determinant of A is 0 means |A| = 0

=>We know that, Caley Hamilton theorem, |A - I| = 0 where is an eigen value of matrix A.

=>When = 0 then |A| = 0 ans when |A| = 0 then we can write it |A - I| = 0 hence = 0

(c)

The answer will be "True"

Explanation:

=>When eigen vectors have distrinct eigen values then corresponding eigen vectors are linearly independent.

(d)

The answer will be "True"

Explanation:

=>Two matrices are called simillar if they have same eigen values, same rank, same polynomial equation or same traces etc.

=>If two matrices are similar then they must have same eigen values and same rank.

=>If two matrices have same eigen values and same rank then they must be similar.

(e)

The answer will be "False"

Explanation:

=>A square matrix A is called diagonalizable if and only if for each eigen value c of A, the the geometric muliplity is less than or equal to the algebric multiplicity.

I have explained each and every part with the help of statements attached to it.