Equation with orthogonal matrix

Question

If $\rm A$ is a $2 \times 2$ orthogonal matrix such that $\mathrm A 1_2 = b \mathrm e_1$, where $\mathrm e_1 = (1,0)$. What is the value of $b$?

I know $A$ being an orthogonal matrix means that $A A^T = I$ but I am unsure how to use that here in this equation?

Answer 1

Since $\bigl\lVert(1,1)\bigr\rVert=\sqrt2$, since $\bigl\lVert(1,0)\bigr\rVert=1$, and since $v\mapsto A.v$ preserves norms, the only possible values for $b$ are $\pm\sqrt2$. Can you take it from here?