Final answer:
Matrix A must have linearly independent rows for Ax=0 to only have the trivial solution, while matrix B must have linearly dependent rows for Bx=0 to have nontrivial solutions.
Step-by-step explanation:
When looking to create matrices A and B such that Ax=0 has only the trivial solution and Bx=0 has a nontrivial solution, you are essentially exploring the concept of matrix equations and solutions to linear systems. For A to have only the trivial solution, it must be a matrix with linearly independent rows, meaning the rank of the matrix is equal to the number of columns. Conversely, for B to have a nontrivial solution, it must have linearly dependent rows which are not all zeroes, resulting in a matrix rank that is less than the number of columns.
An example of matrix A could be:
[[1, 0],
[0, 1],
[0, 0]]
This matrix has full rank (2), meaning any solution to Ax=0 must be the trivial solution, x=0.
Matrix B could be:
[[1, 2],
[2, 4],
[0, 0]]
Here, the second row is a multiple of the first row, indicating linear dependence. There are infinitely many nontrivial solutions to Bx=0 because the system of equations is underdetermined, and thus, there is a free variable.