What are all the possible values for the determinant of an n times n orthogonal

Question

What are all the possible values for the determinant of an n times n orthogonal matrix Q? You can approach this question geometrically, recalling that every orthogonal matrix is a composition of rotations and reflections. You can also approach the question algebraically, starting from the formula Q^T Q = I and using the algebraic properties of determinants. If A is a 5 times 5 matrix and det A = 3, find det (-A) det(2A) det(A^-1) det(A^T A) the rank of A If P is a projection onto a plane in R^3, what is det P?

Explanation / Answer

3.3

d) det (A^t) = det A

hence, det (A^t ) = 3

det (A^tA) = det(A^t)det (A) = 3*3 = 9

c) det (A^-1) = 1/ det(A)

det(A^-1) = 1/ 3

b) det (2A) = 2* det (A) = 2* 3 = 6

a) det (-A) = - det (A)

det (-A) = -3

e) rank of A is 2 since determinant is not equal to 0

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