First slide

Introduction to Determinants

Question

Suppose A and B are two orthogonal matrices of the same size such that $\det (A) + \det (B) = 0$ then

Moderate

A

$A + B = - I$
B

$A + B = I$
C

$\det (A + B) = 0$
D

$A + B = O$

Solution

As A, B are orthogonal matrices, $A A^{'} = B B^{'} = I,$
$\det (A) \neq 0, \det (B) \neq 0 .$ Let $C = A + B$
$C^{'} = A^{'} + B^{'} \Rightarrow A C^{'} = A A^{'} + A B^{'} = I + A B^{'}$
$\Rightarrow A C^{'} B = B + A B^{'} B = B + A$
Now,
$\det (A + B) = \det (A C^{'} B) = \det (A) \det (C^{'}) \det (B)$
$= \det (A) \det (C) (- \det (A))$
$\Rightarrow \det (C) = - (\det (A))^{2} \det (C) = - \det (C)$
$\Rightarrow \det (C) = 0$

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