I'm hoping to show that $|Q| = +1$ or $-1$ if $Q$ is a $p \times p$ orthogonal matrix.
Since I know that $|QQ'| = |I|$ and $|Q||Q'| = |Q|^2$, then $|Q|^2 = |I|$
How should I approach this proof(alternative proof) with the definition of the determinant of the square matrix $k \times k$, such that
$$|A| = a_{11} \quad \text{if }\,k=1$$ $$|A| = \sum_{j=0}^k a_{1j} |A_{1j}| (-1)^{1+j} \quad \text{if }\, k>1$$
where $A_{1j}$ is the $(k-1) \times (k-1)$ matrix obtained by deleting the first row $j$th column of $A$.
Correct me if I'm wrong. If $Q$ = |I|, then $Q = [q_{ij}], Q_{ij} = 0, i\ne j$
$$|Q| = q_{11}Q_{11}+0+...+0 = 1 ?$$