Final answer:
To construct matrix B that results in a zero matrix when multiplied with matrix A, B's columns must be orthogonal to A's rows, ensuring their dot product equals zero.
Step-by-step explanation:
The question pertains to the construction of a 2x2 matrix B such that when multiplied by another matrix A, the resultant product is a zero matrix. In essence, we are tasked with finding a matrix B with two different non-zero columns that, when undergoing matrix multiplication with A, yields the zero matrix. This scenario implies that the columns of B must be created in such a way that they are orthogonal to the rows of A, or in other words, their dot product equals zero.
To construct such a matrix B, take for example matrix A with rows r1 and r2. Matrix B should then have columns c1 and c2 such that:
- r1 · c1 = 0 and r1 · c2 = 0
- r2 · c1 = 0 and r2 · c2 = 0
Each entry in matrix B must be chosen carefully to satisfy the conditions above. While there are multiple possible matrices that can satisfy these conditions, an example is not provided in the question, thus preventing a specific answer. However, a general guideline is to ensure the columns of B are linearly independent and orthogonal to the corresponding rows of A such that their dot products result in zero.