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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?

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    $\begingroup$ This is a great problem. It gives me ideas for my own course. $\endgroup$ – James S. Cook Mar 30 at 17:39
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    $\begingroup$ 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. $\endgroup$ – ErotemeObelus Mar 30 at 18:19
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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?

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  • $\begingroup$ for the second matrix the first column's missing number is -1 and the third column is 1 ,-1 -,1, is there a step by step method to solve these types of problems $\endgroup$ – Samurai Bale Mar 30 at 18:02
  • $\begingroup$ The method that I described is step-by-step. $\endgroup$ – José Carlos Santos Mar 30 at 18:04

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