Linear Algebra Prove that if A is orthogonal, than A^T is orthogona l I am not s

Question

Linear Algebra

Prove that if A is orthogonal, than A^T is orthogonal

I am not sure if I approached my proof precisely. Let me know if there is a better way or if I need to change something on the one I did. It needs to be simple to remember for a test please :)

My attempt using the identity "I" and the determinant +/-1:

A^TA=I

det(A^TA)=det(I)

det(A^T)det(A)=det(I)

det(A^T)det(A)=1

det(A)det(A)=1

(det(A)^2)=1

sqrt(det(A))^2=+/- sqrt(1)

det(A) = +/- 1 = det(A^T)

since, det(A) = det(A^T)

Help

Thanks

Explanation / Answer

Yes, yours is correct but there is a much much simpler proof

matrix "A" is orthogonal if A.(A^T)=I

Now Let B=A^T

A^T =B is orthogonal if B.(B^T)=I

= (A^T)(A^T^T) (Since A^T^T=A)

So, =(A^T)(A)

Since it is already mentioned that A.(A^T)=I hence (A^T)A is also I and hence A^T is also orthogonal

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