Question

Construct a 4 x 3 matrix A with rank 1. Select the correct choice and, if necessary, fill in the answer box to complete your choice.

A. A=
B. There is no 4x3 matrix with rank 1.

Answer 1

A 4 x 3 matrix with rank 1 can be constructed by ensuring all rows are scalar multiples of one non-zero row. An example is a matrix with rows [2, 3, 4], [4, 6, 8], [6, 9, 12], and [8, 12, 16].

Step-by-step explanation:

To construct a 4 x 3 matrix A with rank 1, we need to ensure that all rows of the matrix are linearly dependent on each other, that is, they must be scalar multiples of each other. Here is an example of how you can construct such a matrix:

Let's pick a non-zero row, for instance, [2, 3, 4]. Since the matrix has rank 1, all other rows must be scalar multiples of this row. Thus, the matrix A could be:

A =
[
[2, 3, 4],
[4, 6, 8], // 2 times the first row
[6, 9, 12], // 3 times the first row
[8, 12, 16] // 4 times the first row
]

This matrix A has rank 1 because all of its rows are multiples of the first row. Consequently, there is only one linearly independent row, which defines the rank of the matrix.