Orthogonal Matrix

[helenli89]

Prove that if R is an orthogonal matrix, then det(R) = +/- 1.

I know R is orthogonal if and only if the columns of R form an orthonormal set and the rows of R form an orthonormal set. Which means that if R = (r1 r2; r3 r4) an M(2,2) matrix where the Row1=r1 r2 and Row2=r3 r4. Then r1*r3+r2*r4 = r1*r2+r3*r4 =0 and that the Determinat of R is det(R) = r1*r4-r2*r3
But how do I collect them together to show that det(R)=1 or -1? I have tried to do the following:
since r1r2+r3r4=0=r1r3+r2r4. so: r1r2+r3r4=r1r3+r2r4. Then: r1r2-r1r3=r2r4-r3r4 -> r1(r2-r3)=r4(r2-r3) -> r1=r4
Similarly, r3=r2. So the det(R) = r1r4-r2r3 = r1^2-r2^2 = r4^2-r3^2
But then how do I prove that r1^2-r2^2 = r4^2-r3^2 =1or -1?

Tanks

 

[tkhunny]

You seem to be using only half the available information. Check that definition of Orthogonal Matrices.

Orthogonal R implies \(\displaystyle R^{T}\;\cdot\;R\;=\;I\).

For R a 2x2 matrix, that's four chunks of information.

 

[helenli89]

You seem to be using only half the available information. Check that definition of Orthogonal Matrices.

Orthogonal R implies \(\displaystyle R^{T}\;\cdot\;R\;=\;I\).

For R a 2x2 matrix, that's four chunks of information.

I know that R^T * R = I. But how can just writting "because R^T * R = I therefore (det(R))^2 = det(R) det(R^T) = det (R*R^T) = det (I) = 1" be enough for a prove and where does the -1 come from?

 

[tkhunny]

You seem to have missed the point.

With your matrix, you have:

\(\displaystyle r1^{2} + r2^{2} = 1\)

\(\displaystyle r1 \cdot r3 + r2 \cdot r4 = 0\)

\(\displaystyle r3^{2} + r4^{2} = 1\)

This leads to the inevitable conclusions that:

r1 = r4 = 0

r2 = +1 or -1

r3 = +1 or -1

Without additional information, that is all that can be said.

What is the implication for your determinant?