Find orthogonal matrix missing values I was solving this problem, where I need to find the value x, which is missed in the orthogonal matrix A.
$$
A =
\begin{pmatrix}
x&amp;0.5&amp;-0.5&amp;-0.5\\
x&amp;0.5&amp;0.5&amp;0.5\\
x&amp;-0.5&amp;-0.5&amp;0.5\\
x&amp;-0.5&amp;0.5&amp;-0.5\\
\end{pmatrix}
$$
One of the properties of orthogonal matrix is that the dot product of orthogonal matrix and its transposed version is the identity matrix:
$$I=A.A^T$$
$$
A^T =
\begin{pmatrix}
x&amp;x&amp;x&amp;x\\
0.5&amp;0.5&amp;-0.5&amp;-0.5\\
-0.5&amp;0.5&amp;-0.5&amp;0.5\\
-0.5&amp;0.5&amp;0.5&amp;-0.5\\
\end{pmatrix}
$$
So I figured that $$x=0.5$$
That would result in identity matrix I:
$$
I =
\begin{pmatrix}
1&amp;0&amp;0&amp;0\\
0&amp;1&amp;0&amp;0\\
0&amp;0&amp;1&amp;0\\
0&amp;0&amp;0&amp;1\\
\end{pmatrix}
$$
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!
$A$ is orthogonal $ \iff x= \pm \frac{1}{2}.$
Thank you everyone, indeed there are two possible answers $$x = 1/2$$ $$x = -1/2$$, therefore there can't be a definitive answer.