Corollary. The character table of a group is an invertible square matrix.
The theorem that is a corollary to states that the character table is a square matrix and the explanation for invertibility given is the that rows are orthogonal with respect to $\langle \ , \ \rangle$.
I cannot understand why orthogonality of the rows is enough to show invertibility.
Since the matrix is square, it is enough to show that the rows are linearly independent. But since they are orthogonal with respect to an inner product, and each has inner product $1$ with itself, this is a standard exercise in linear algebra (we just need that for each row $v$ we have $\langle v,v\rangle\neq 0$).
To elaborate a bit on the exercise, one considers a linear combination $a_1v_1+\cdots + a_nv_n = 0$ and takes inner product with each $v_i$ in turn to see that the coefficient $a_i$ must be $0$.