 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2023/digital-systems-linear-algebra-and-introduction-to-formal-proof/4/f/solution

Expand the product and use (d) together with its transpose:

M M^{⊤} = (A + B) (A^{⊤} + B^{⊤}) = A A^{⊤} + A B^{⊤} + B A^{⊤} + B B^{⊤} = A A^{⊤} + B B^{⊤} .

Indeed

A B^{⊤} = 0

and

B A^{⊤} = (A B^{⊤})^{⊤} = 0

 New to topics? Read the docs here!