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IfA=[[ 1 2 3;...

Question

If $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 4 & 5 & 6 7 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦, B = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 3 & 0 0 & 4 & 5 \end{matrix} ⎤ ⎥ ⎦$ and $T r (A B) = λ T r (A) \cdot T r (B)$ , then $λ$ is equal to

A

1

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B

\frac{27}{20}

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C

\frac{20}{27}

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D

\frac{1}{2}

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Solution

The correct option is C $\frac{20}{27}$
Given, $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 4 & 5 & 6 7 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦ T r (A) = 1 + 5 + 0 = 6 B = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 3 & 0 0 & 4 & 5 \end{matrix} ⎤ ⎥ ⎦ T r (B) = 1 + 3 + 5 = 9$
Now, $A B = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 4 & 5 & 6 7 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 3 & 0 0 & 4 & 5 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow A B = ⎡ ⎢ ⎣ \begin{matrix} 1 & 18 & 15 4 & 39 & 30 7 & 3 & 0 \end{matrix} ⎤ ⎥ ⎦$ $T r (A B) = 1 + 39 + 0 = 40$
Also, given $T r (A B) = λ T r (A) \cdot T r (B)$ $\Rightarrow λ = \frac{40}{54} = \frac{20}{27}$

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