# 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

Consider the three following matrices:

Calculate the Tr(ABC)
(a)1
(b)2
(c)2i
(d)0
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Elberte
Step 1
The given matrices are

Step 2
Finding the product ABC.
$ABC=\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& -1\end{array}\right]\left[\begin{array}{ccc}0& -i& 0\\ i& 0& -i\\ 0& i& 0\end{array}\right]$
$=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& -1\end{array}\right]\left[\begin{array}{ccc}0& -i& 0\\ i& 0& -i\\ 0& i& 0\end{array}\right]$
$=\left[\begin{array}{ccc}0& -i& 0\\ 0& 0& 0\\ 0& -i& 0\end{array}\right]$
Hence,
$ABC=\left[\begin{array}{ccc}0& -i& 0\\ 0& 0& 0\\ 0& -i& 0\end{array}\right]$
Step 3
Tr(ABC)=sum of the diagonal enteries of the matrix ABC
Tr(ABC)=0+0+0
$⇒Tr\left(ABC\right)=0$
Hence, option d is correct.
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Jeffrey Jordon

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