Orthogonal Matrices

Algebra Level 5

( 0.3 b c l m n o p q ) \large \begin{pmatrix} 0.3 & b & c \\ l & m & n \\ o & p & q \end{pmatrix}

If the matrix above is orthogonal, find the sum of squares of all the possible values of 10 ( m q n p ) 10(mq-np) .


The answer is 18.

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2 solutions

Satvik Golechha
May 25, 2017

Since it is an orthogonal matrix, we know, A T = A 1 = a d j ( A ) A A^T=A^{-1}=\frac{adj(A)}{|A|} . Since the two matrices are equal, and the value of determinant of an orthogonal matrix is ± 1 \pm 1 , we can equate the first term to get 10 ( m q n p ) = ± 3 10(mq-np)=\pm 3 . The sum of their squares is 18 18 .

Is there an intuitive / non-calculation reason for why m q n p = ± a mq - np = \pm a ?

Calvin Lin Staff - 4 years ago
Kszir Kiatsow
Aug 1, 2019

I googled determinant of orthogonal matrix and found https://math.stackexchange.com/questions/122639/sub-determinants-of-an-orthogonal-matrix, from which I concluded that the determinant of the bottom right 2 by 2 submatrix can be either 0.3 or -0.3.

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