Question

# Consider the three following matrices: A=begin{bmatrix}1 & 0&0 0 & -1&00&0&1 end{bmatrix} , B=begin{bmatrix}1 & 0&0 0 & 0&00&0&-1 end{bmatrix} text{ and } C=begin{bmatrix}0 & -i&0 i & 0&-i0&i&0 end{bmatrix} Calculate the Tr(ABC) (a)1 (b)2 (c)2i (d)0

Matrices
Consider the three following matrices:
$$A=\begin{bmatrix}1 & 0&0 \\0 & -1&0\\0&0&1 \end{bmatrix} , B=\begin{bmatrix}1 & 0&0 \\0 & 0&0\\0&0&-1 \end{bmatrix} \text{ and } C=\begin{bmatrix}0 & -i&0 \\i & 0&-i\\0&i&0 \end{bmatrix}$$
Calculate the Tr(ABC)
(a)1
(b)2
(c)2i
(d)0

2020-10-29
Step 1
The given matrices are
$$A=\begin{bmatrix}1 & 0&0 \\0 & -1&0\\0&0&1 \end{bmatrix} , B=\begin{bmatrix}1 & 0&0 \\0 & 0&0\\0&0&-1 \end{bmatrix} \text{ and } C=\begin{bmatrix}0 & -i&0 \\i & 0&-i\\0&i&0 \end{bmatrix}$$
Step 2
Finding the product ABC.
$$ABC=\begin{bmatrix}1 & 0&0 \\0 & -1&0\\0&0&1 \end{bmatrix}\begin{bmatrix}1 & 0&0 \\0 & 0&0\\0&0&-1 \end{bmatrix}\begin{bmatrix}0 & -i&0 \\i & 0&-i\\0&i&0 \end{bmatrix}$$
$$=\begin{bmatrix}1 & 0&0 \\0 & 0&0\\0&0&-1 \end{bmatrix}\begin{bmatrix}0 & -i&0 \\i & 0&-i\\0&i&0 \end{bmatrix}$$
$$=\begin{bmatrix}0 & -i&0 \\0 & 0&0\\0&-i&0 \end{bmatrix}$$
Hence,
$$ABC=\begin{bmatrix}0 & -i&0 \\0 & 0&0\\0&-i&0 \end{bmatrix}$$
Step 3
Tr(ABC)=sum of the diagonal enteries of the matrix ABC
Tr(ABC)=0+0+0
$$\Rightarrow Tr(ABC)=0$$
Hence, option 'd' is correct.