Question

Suppose the last column of AB is entirely zero but B itself has no column of zeros. What can you say about the columns of​ A? Choose the correct answer below. A. The columns of A must be the identity matrix columns because if the last column in B is denoted ​, then the last column of AB can be rewritten as A0. This implies that A must be the identity matrix to ensure that the last column of AB is all zeros. B. The columns of A are linearly independent because if the last column in B is denoted ​, then the last column of AB can be rewritten as A0. Since is not all​ zeros, then the only solution to A0 is the trivial solution. C. The columns of A are linearly dependent because if the last column in B is denoted ​, then the last column of AB can be rewritten as A0. Since is not all​ zeros, then any solution to A0 can not be the trivial solution. D. The columns of A must all have entries of zero because if the last column in B is denoted ​, then the last column of AB can be rewritten as A0. This implies that A must have columns that are composed of only zeros and therefore AB must be the zero matrix.

Answer 1

The correct option is C, the columns of A are linearly dependent because if the last column in B is denoted ​, then the last column of AB can be rewritten as A0. Since B is not all​ zeros, then any solution to A0 can not be the trivial solution

Explanation:

Firstly, have to define linear dependency to get a clear picture of what is to be done.

If we have two vectors A and B, they are said to be linearly independent if Ax + By = 0 if and only if x = y = 0. Otherwise they are linearly dependent.

Now, let p denote the last column of the matrix B. We are given that the last column of AB is zero, therefore Ap = 0. Although p is not the zero vector, because B has no columns of zeros, therefore the equation Ap = 0 is a linear dependence relation among the columns of A. Hence, the columns of A are linearly dependent.