Question

How to determine `a` and `b` so that the real matrix `((a,1,b),(b,a,1),(1,b,a))` is orthogonal matrix ?

Answer 1

`a=b=0`

Explanation:

A matrix `M` is orthogonal if and only if:

`MM^T = M^TM = I`

Evaluating this directly with the given form of matrix:

`((a,1,b),(b,a,1),(1,b,a))((a,1,b),(b,a,1),(1,b,a))^T`

`= ((a,1,b),(b,a,1),(1,b,a))((a,b,1),(1,a,b),(b,1,a))`

`=((a^2+b^2+1,ab+a+b,ab+a+b),(ab+a+b,a^2+b^2+1,ab+a+b),(ab+a+b,ab+a+b,a^2+b^2+1))`

In order for this to be the identity matrix, we require:

`a^2+b^2+1 = 1`

So if `a`, `b` are real then they must both be `0`.

Then we find:

`ab+a+b=0`

satisfying the requirement that the off diagonal elements be `0`.

Answer 2

See below

Explanation:

Also, an orthogonal matrix has columns and rows that are orthogonal unit vectors.

For the first column vector:

`abs(mathbfv_1) = sqrt (a^2 + b^2 + 1^2) = 1 implies a = b = 0 " if " a,b in RR`.