3 Let A be a 7 7 with rankA 1 a Explain why 0 is an eigenv

3. Let A be a 7 × 7 with rank(A) = 1.
(a) Explain why 0 is an eigenvalue of A and find gemu(0).
(b) Suppose that A has an additonal eigenvalue * /= 0. Show that A is diagonalizable.

Solution

Solution: (a) Given that A is a 7x7 matrix with rank =1.

So A is singular matrix, since 1<7.

Therefore determinant(A) = 0.

Since product of eigenvalues gives its detenminant with its sign positive or negative

according to n(order of matrix) even or odd, atleast one of the eigenvalue must be zero(0).

(b) if A has all distinct eigenvalues then their corresponding eigenvectors must be linearly independent

and A will be diagonalizable.