Problem 2. (4 points) Suppose A is a matrix of size 4 by 4. Which of the following statements must be TRUE? (I) If the rank of A is 4, then the matrix A must be invertible. (II) If the matrix A is invertible, then the rank of A is 4. (III) If A is invertible, then the nullity of A is 0. (A) I only (B) II only (C) III only (D) II and III only (E) I, II, and III

(E) I, II, and III

Step-by-step explanation:

Suppose the matrix  A has rank 4.

A has 4 linearly independent columns.

As the matrix  A is 4 by 4 matrix so all columns of A are linearly independent.

=> det(A) ≠ 0.

=> A must be invertible.

Suppose A is invertible.

Columns of A are linearly independent.

As A has 4 columns and all columns of A are linearly independent so A has 4 linearly independent columns.

As Rank of A = Number of linearly independent columns of A.

=> Rank of A = 4.

Suppose A is invertible.

=> Rank of A = 4.

By rank nullity theorem,

Rank of A + Nullity of A= 4

=> 4 + Nullity of A= 4

=> Nullity of A= 0.

Hence the nullity of A is 0.

ian gonna lie that is way to much