⎝ ⎜ ⎛ 0 . 3 l o b m p c n q ⎠ ⎟ ⎞
If the matrix above is orthogonal, find the sum of squares of all the possible values of 1 0 ( m q − n p ) .
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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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Since it is an orthogonal matrix, we know, A T = A − 1 = ∣ A ∣ a d j ( A ) . Since the two matrices are equal, and the value of determinant of an orthogonal matrix is ± 1 , we can equate the first term to get 1 0 ( m q − n p ) = ± 3 . The sum of their squares is 1 8 .