Question

Consider the three following matrices:
A (1 0 0
0 -1 0
0 0 1)
B =( 1 0 0
0 0 0
0 0 -1 )
C = ( 0 -i 0
i 0 -i
0 i 0 )
Calculate the Tr( (ABC) .
(a) (b) 2 (c) 2i (d) 0 ​

Answer 1

Answer:

D

Explanation:

Step 1

The given matrices are

A=⎡⎣⎢1000−10001⎤⎦⎥,B=⎡⎣⎢10000000−1⎤⎦⎥ and C=⎡⎣⎢0i0−i0i0−i0⎤⎦⎥

Step 2

Finding the product ABC.

ABC=⎡⎣⎢1000−10001⎤⎦⎥⎡⎣⎢10000000−1⎤⎦⎥⎡⎣⎢0i0−i0i0−i0⎤⎦⎥

=⎡⎣⎢10000000−1⎤⎦⎥⎡⎣⎢0i0−i0i0−i0⎤⎦⎥

=⎡⎣⎢000−i0−i000⎤⎦⎥

Hence,

ABC=⎡⎣⎢000−i0−i000⎤⎦⎥

Step 3

Tr(ABC)=sum of the diagonal enteries of the matrix ABC

Tr(ABC)=0+0+0

⇒Tr(ABC)=0

Hence, option 'd' is correct