Find orthogonal matrix missing values

Question

I was solving this problem, where I need to find the value x, which is missed in the orthogonal matrix A.

$$ A = \begin{pmatrix} x&0.5&-0.5&-0.5\\ x&0.5&0.5&0.5\\ x&-0.5&-0.5&0.5\\ x&-0.5&0.5&-0.5\\ \end{pmatrix} $$

One of the properties of orthogonal matrix is that the dot product of orthogonal matrix and its transposed version is the identity matrix:

$$I=A.A^T$$

$$ A^T = \begin{pmatrix} x&x&x&x\\ 0.5&0.5&-0.5&-0.5\\ -0.5&0.5&-0.5&0.5\\ -0.5&0.5&0.5&-0.5\\ \end{pmatrix} $$

So I figured that $$x=0.5$$

That would result in identity matrix I:

$$ I = \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{pmatrix} $$

As the result I had to provide the sum of missing values, which $0.5\times4 = 2$. The grader (program) returned the error. Is there something I am missing (there most likely is, I guess)?

Thank you in advance!

Answer 1

$A$ is orthogonal $ \iff x= \pm \frac{1}{2}.$

Answer 2

There are two solutions: $\frac12$ and $-\frac12$.

Answer 3

You don't say how you found $x$. If you actually compute $A^TA$ and compare with $I$, you get that $|x|=1/2$, so $x$ is either $1/2$ or $-1/2$.

Answer 4

A short test is that $\pm 1=\det(A)=-2x$. This leaves both possibilities, namely $+1$ and $-1$ for the determinant, so $x=1/2$ and $x=-1/2$. Indeed, one checks that both cases yield an orthogonal matrix.

Answer 5

Thank you everyone, indeed there are two possible answers $$x = 1/2$$ $$x = -1/2$$, therefore there can't be a definitive answer.