First slide

Transpose of matrix

Question

If A is a $2 \times 2$ unitary matrix, then |A| is equal to

Moderate

A

1
B

-1
C

$\pm 1$
D

none of these

Solution

Let $A = [\begin{array}{ll} a b \\ c d \end{array}], A^{} = [\begin{array}{cc} \bar{a} & \bar{c} \\ \bar{b} & \bar{d} \end{array}]$
$\begin{aligned} A A^{} = I \Rightarrow | A | |A^{*}| = 1 \\ \Rightarrow (a d - b c) (\bar{a} \bar{d} - \bar{b} \bar{c}) = 1 \\ \Rightarrow | a d - b c |^{2} = 1 \end{aligned}$
$\Rightarrow a d - b c = e^{i θ}$ for some $θ \in R$

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