Let A and B be two n x n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.
I. rank(AB) = rank(A) rank(B)
II. det(AB) = det(A) det(B)
III. rank(A+B) ≤ rank(A) + rank(B)
IV. det(A+B) ≤ det(A) + det(B)
Which of the above statements is TRUE?
A.
I and II only
B.
I and IV only
C.
II and III only
D.
III and IV only
Solution:
Given:
rank(M)= rank of the matrix
det(M)= determinant of the Matrix
We know, rank(AB) = minimum(rank(A), rank(B))
det(AB) = det(A) × det(B)
rank(A+B) ≤ rank(A) + rank(B)
Since the addition of two matrices can never result in an increase in the number of independent columns and rows in the matrix.
Hence, the correct option is C.