Question-65
Let \(\displaystyle A,B,C\) be three square matrices. Which of the following is equal to \(\displaystyle \text{tr}( ABC)\)?
Use the property that \(\text{tr}(AB) = \text{tr}(BA)\)
We have:
\[ \begin{equation*} \begin{aligned} \text{tr}( ABC) & =\text{tr}( A( BC))\\ & =\text{tr}(( BC) A)\\ & =\text{tr}( BCA) \end{aligned} \end{equation*} \]
Similarly:
\[ \begin{equation*} \begin{aligned} \text{tr}( ABC) & =\text{tr}(( AB) C)\\ & =\text{tr}( C( AB))\\ & =\text{tr}( CAB) \end{aligned} \end{equation*} \]
As a counter example to options (3) and (4), consider the matrices:
\[ \begin{equation*} A=\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} ,\ B=\begin{bmatrix} 0 & 1\\ 0 & 0 \end{bmatrix} ,\ C=\begin{bmatrix} 0 & 0\\ 1 & 0 \end{bmatrix} \end{equation*} \]
We see that \(\displaystyle \text{tr}( BAC) =\text{tr}( ACB) =0\).