Chapter 5

Trace of a (Square) Matrix

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Problem 5.1

Show that for any $m \times n$ matrix $A$ , $n \times p$ matrix $B$ , and $p \times q$ matrix $C$ ,

\mathrm{tr}(ABC) = \mathrm{tr}(B'A'C') = \mathrm{tr}(A'C'B').

Problem 5.2

(a)(b)

Let $A$ , $B$ , and $C$ represent $n \times n$ matrices.

(a)

Using the result of Exercise 5.1 (or otherwise), show that if $A$ , $B$ , and $C$ are symmetric, then $\mathrm{tr}(ABC) = \mathrm{tr}(BAC)$ .

(b)

Show that [aside from special cases like that considered in Part (a)] $\mathrm{tr}(BAC)$ is not necessarily equal to $\mathrm{tr}(ABC)$ .

Problem 5.3

(a)(b)

Let $A$ represent an $n \times n$ matrix such that $A'A = A^2$ .

(a)

Show that $\mathrm{tr}[(A-A')'(A-A')] = 0$ .

(b)

Show that $A$ is symmetric.