# solving for scalar involving matrix equations

consider following matrix equation

$$\boldsymbol{1}^T \Sigma^{-1} ( \bar{\textbf{x}} - n \boldsymbol{1}. c) = 0$$

here $$\boldsymbol{1} , \bar{\textbf{x}}$$ are vectors of dimension $$n \times 1$$ and $$\Sigma$$ is invertible matrix of dimension $$n \times n$$ and n,c are constants

is there any way to solve it for scalar c ?

$$\boldsymbol{1}^T \Sigma^{-1}\bar{\textbf{x}} = (n.c)\boldsymbol{1}^T \Sigma^{-1}\boldsymbol{1}$$

and I am struck at here

$$c = \frac{ \boldsymbol{1}^T \Sigma^{-1}\bar{\textbf{x}} }{ n \boldsymbol{1}^T \Sigma^{-1}\boldsymbol{1} }.$$
Remember that $$\boldsymbol{1}^T \Sigma^{-1}\boldsymbol{1}$$ and $$\boldsymbol{1}^T \Sigma^{-1}\bar{\textbf{x}}$$ are scalars.