Final answer:
To build a 2x2 matrix B such that AB results in the zero matrix, B's columns must be non-trivial, nonzero vectors that lie in the null space of A. By solving the homogeneous equations defined by A, distinct solutions can be used as columns for B.
Step-by-step explanation:
To construct a 2x2 matrix B such that the product of matrix A and B is the zero matrix, we must identify vectors which when multiplied by A, result in a zero vector. Let's assume that matrix A has two linearly independent rows. Then, we are looking for two non-trivial columns for B which are in the null space of A.
For example, if A is the matrix:
[ c d ]
We could choose B with columns that are solutions to the homogeneous equations:
- a*x + b*y = 0
- c*x + d*y = 0
As long as we choose two distinct (nonzero) solutions for (x, y), we can ensure that AB will be the zero matrix. Note that it's crucial for these solutions to be non-trivial (nonzero) and distinct to satisfy the student's conditions.
Here's an example for a specific matrix, A = [[1,2],[3,4]]:
The homogeneous system is:
- 1*x + 2*y = 0
- 3*x + 4*y = 0
One might select x = -2, y = 1 for the first column and x = -4, y = 1 for the second column, yielding matrix B = [[-2, -4],[1, 1]] as an appropriate choice.