Question

Problem 4. (3 points) Given the following matrix A = 1 1 0 1 . Which of the following statements must be TRUE? (I) The rank of the matrix is one. (II) The matrix A is not diagonalizable. (III) The nullity of A is one. (A) I only (B) II only (C) III only (D) II and III only (E) I, II, and III

Answer 1

C) II only

Explanation:

Given the 2×2 matrix A = [1 1, 0 1]

The only statement true there is (II) only i.e the matrix is not diagonalizable but invertible. A matrix is invertible if the product of the matrix and its inverse is equal to an identity matrix

The rank of the matrix is not 1 but 2 because rank of a matrix is the number of non zero rows of a matrix and the number of non zero rows of this matrix is 2 thereby making I incorrect.

Also note that the sum of nullity and rank is equal to the number of columns of the given matrix

Nullity + rank = number of columns

Nullity = number of columns - rank

Nullity = 2-2 = 0

The nullity is therefore 0 not 1 making option III also false