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

Let A be a 7 × 7 with rank(A) = 1.

(a) Explain why 0 is an eigenvalue of A and find gemu(0).

Solution

(a) Since rank of A is 1 and A is a matrix of order 7, so rank of A is less than order of matrix. In this case A is always singular because if A is non-singular, then rank should be 7.

If A is a singular matrix, then det(A)=0 or det(A-0I)=0 this means that 0 is an eigenvalue of A.

Since order of A is 7 and rank of A is 1, so the geometric multiplicity of a is 7-1=6