Final answer:
The question relates to the properties determined by the given determinants of the matrices A and B, and potential calculations involving these matrices under operations like addition and multiplication. It could also extend to relations between vectors using their dot product and cross product, indicating their directions and orthogonality.
Step-by-step explanation:
If we are given that det (A) = 2 and det (B) = -5 for 4×4 matrices A and B, you might be wanting to find properties related to these determinants, such as the determinant of their product or their linear combinations.
In the context of the given information regarding cross product, dot product, and vector relations, it seems that the specific questions might involve calculating the determinant of a matrix resulting from these operations, or understanding the relationship between vectors which would require using the properties of cross product and dot product.
For instance, if you are interested in the determinant of the matrix C, given by C+A=2B, you would use the fact that the determinant is a multiplicative map to calculate it based on the known determinants of A and B. Similarly, if you are given that A x B represents the cross product of vectors, and you are looking for the properties of these vectors based on their cross product being zero or their dot product being zero, you would use the geometric properties of vectors like orthogonality and parallelism.
For two vectors, if their cross product vanishes, it means they are parallel or one is the zero vector. If their dot product vanishes, the vectors are orthogonal (perpendicular). The dot product of a vector with the cross product of that vector with another is always zero since it represents the volume of a parallelepiped with height zero or because they are orthogonal in space.