Inverse of a special matrix

Question

Is there easy (analytical) way to find the inverse of the following matrix, where $C$ is a vector? $$ \begin{bmatrix} 1 & C^\top \\ C & CC^\top \end{bmatrix} $$

Answer 1

The matrix has no inverse, unless $C$ is the null vector and the matrix is $[1]$.

The matrix is an outer product of the vector $[1\quad C^T]$. All outer product matrices have rank one, meaning it is rank deficient.

For instance, let $C^T=[x,y]$ then $$ CC^T=\left[\begin{array}{cc}x^2&xy\\xy&y^2\end{array}\right] $$ and $$ \left[\begin{array}{cc}1&C^T\\C&CC^T\end{array}\right]=\left[\begin{array}{ccc}1&x&y\\x&x^2&xy\\y&xy&y^2\end{array}\right] $$ In which the columns are clearly all multiples of one another.

Answer 2

Such a matrix cannot be invertible, since

$$\begin{bmatrix} 1 & C^\top \\ C & CC^\top \end{bmatrix} =\begin{bmatrix} 1 \\ C \end{bmatrix} \begin{bmatrix} 1 & C^\top \end{bmatrix} =\begin{bmatrix} 1 \\ C \end{bmatrix}\begin{bmatrix} 1 \\ C \end{bmatrix} ^\top$$ is rank one and the dimensions are larger than one.