midtlinjeg

## Answered question

2020-10-28

Consider the three following matrices:

Calculate the Tr(ABC)
(a)1
(b)2
(c)2i
(d)0

### Answer & Explanation

Elberte

Skilled2020-10-29Added 95 answers

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.

Jeffrey Jordon

Expert2022-01-30Added 2605 answers

Answer is given below (on video)

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