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

Question

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?

Answer 1

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?

Answer 2

Since you ask for a step-by-step method explicitly in a comment, here it is:

1. Replace blanks with variables in the matrix. For example: $$A=\begin{bmatrix}-1&1&-1&1\\1&1&-1&a\\0&1&b&c\\0&0&1&d\end{bmatrix}$$
2. Now, for every pair of distinct columns, form the equation corresponding to their dot product equating zero. For example, for the second and fourth columns the equation is: $$ 1 \cdot 1 + 1 \cdot a + 1 \cdot c + 0 \cdot d= 0 $$ Do this for all column pairs. This will give you a system of equations (not necessarily a linear system).
3. Now you try to solve this system of equations for values of the variables. Best case, you'll get a single value for each variable.
   However, it might turn out that the system has no solutions, which would mean "it is not possinble to make the matrix columns orthogonal".
   Or, it might turn out that there are several solutions (might even be infinite in number), in which case any of them would work.