# How do I complete a matrix so that its columns are orthogonal?

I'm not sure how to start on these types of problems. I know that the dot product of the columns have to be zero between each pair.

Do I set up a system of linear equations for this? If so, what would it even look like?

• This is a great problem. It gives me ideas for my own course. – James S. Cook Mar 30 at 17:39
• Remember that two columns are orthogonal if and only if their dot product is zero. For matrix A what you need to do is find a, b, c, d so that the dot product is zero pairwise for all columns. And something similar for matrix B. – ErotemeObelus Mar 30 at 18:19

Your matrix $$A$$ is$$A=\begin{bmatrix}-1&1&-1&1\\1&1&-1&a\\0&1&b&c\\0&0&1&d\end{bmatrix}.$$The first to columns are clearly orthogonal. Also, the first and the third columns are orthogonal too, no matter what $$B$$ is. On the other hand, the second and the second columns are orthogonal if and only if $$1\times(-1)+1\times(-1)+1\times b+0\times1=0$$, which means that $$b=2$$. Let us then assume that $$b=2$$
Now, the first and the fourth columns are orthogonal if and only if $$a=1$$. So let us assume that $$a=1$$. Then the second and the fourth columns are orthogonal if and only if $$c=-2$$. Now, complete the process in order to determine the only possible value for $$d$$. Can you deal with the other matrix now?