Suppose A and B are n × n matrices. Which of the following statements is/are true?
Suppose A and B are n × n matrices. Which of the following statements is/are true?
A. (A + B)^T = B^T + A^T
B. det (AB)^T = det (A) . det (B)
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Verified answer
Both are true.
Property A can be checked componentwise. For any 1 <= i, k <= n,
[(A+B)^T]_{ik} = (A+B)_{ki}
= A_{ki} + B_{ki}
= (A^T)_{ik} + (B^T)_{ik}
= (A^T + B^T)_{ik}.
Property B follows from the identities det (AB) = (det A) (det B) and det M^T = det M:
det (AB)^T = det (AB) = (det A) (det B).