Let $X$ be $n\times p$ matrix as $X=(x_1, x_2, \ldots x_p)$. I partition the matrix as follows $X=(X_1, X_2)$ where $X_1$ is a $n\times p_1$ matrix and $X_2$ is a $n\times (p-p_1)$ matrix. Then how can I show the following?
$|I+X X^{\prime}|=|I+X_1 X_1^{\prime} +X_2 X_2^{\prime}|$, where $|A|$ is the determinant of the matrix $A$ and $I$ is the identity matrix of order $n$.
Thanks in advance.