Question

Note: - You are attempting question 5 out of 12 Let A be unitary matrix. For what value of k, kA is also unitary? (a) \k\=1 (b) \k\ = 2 (c) \k = 3 (d) \k} = 4 Answer A B С NG 20DLAAdvanced.png Address

Answer 1

is unitary matrix . For what value of   ,   is also unitary ?

Solution :

Result :

1) Determinant of unitary matrix is  $\pm 1$ .

and

2)
det(k A) = k” det(A)
, where   $A \in M_n(\mathbb{C})$ .

Therefore given     is unitary matrix .

$det(A)=\pm 1$ .

Also if     is also unitary   

then   $det(kA)=\pm 1$

Now
det(k A) = k” det(A)
  $\implies$   $\pm 1 = k^n (\pm 1)$

$\implies |\pm 1 | =|k^n| |\pm1| \implies 1=|k|^n1 \implies |k|=1$

Therefore

$\large |k|=1$

Therefore answer is :

Option a)   $\large |k|=1$