Suppose A and B are n × n matrices with real entries. Which of the follow are TRUE? I. The trace of A^2 is nonnegative. II. If A^2 = A, then the trace of A is nonnegative. III. The trace of AB is the product of the traces of A and B.

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∻Solution by Steps∻ ‖ step 1 ⋮ Review the properties of matrix trace from asksia-ll calculation list. ‖ ‖ step 2 ⋮ The trace of a square matrix $ A $, denoted $ \text{Tr}(A) $, is the sum of its diagonal elements. ‖ ‖ step 3 ⋮ For statement I, the trace of $ A^2 $ is nonnegative if $ A $ is a real symmetric matrix, since the eigenvalues of $ A $ are real and the trace is the sum of the eigenvalues. However, for a general real matrix, this is not necessarily true. ‖ ‖ step 4 ⋮ For statement II, if $ A^2 = A $ (A is idempotent), then the eigenvalues of $ A $ are either 0 or 1, and thus the trace, being the sum of eigenvalues, is nonnegative. ‖ ‖ step 5 ⋮ For statement III, the trace of $ AB $ is generally not equal to the product of the traces of $ A $ and $ B $. This is true only for special cases, such as when $ A $ and $ B $ commute and are simultaneously diagonalizable. ‖ ∻Answer∻ ⚹ Statement II is TRUE; Statements I and III are generally FALSE. ⚹ ∻Key Concept∻ ⚹Trace of a Matrix and its Properties⚹ ∻Explanation∻ ⚹The trace of a matrix is the sum of its diagonal elements. The trace of $ A^2 $ is nonnegative only for certain matrices like real symmetric matrices. If $ A^2 = A $, the matrix is idempotent, and its trace is nonnegative. The trace of $ AB $ is not generally the product of the traces of $ A $ and $ B $.⚹◊What is the formula for the trace of a matrix?⍭ Generate me a similar question◊

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