If the determinant of the matrix of coefficients of a system

If the determinant of the matrix of coefficients of a system of n linear equations in n unknowns is zero, then the rank of the matrix of coefficients is less than n.

(a) Always true

(b) Sometimes true

(c) Never true, i.e., false

(d) None of the above

Solution

Let, A be the matrix

det(A)=0

HEnce, A has eigenvalue 0 so there is non trival solution to Ax=0

ie null(A) is non zero

So, nullity (A)>=1

rank(A)+nullity(A)=n

Hence, rank(A)<n

So,

a) Always true