Suppose that X X is a sample, Extra left or missing ight Extra left or missing ight

Suppose that $X$ is a sample, $\text{Extra left or missing ight}$ is the distance matrix of the sample, $\hat{A}$ is the double-centered distance matrix and $\tilde{A}$ is the U-centered distance matrix of the sample.

(a) State and prove necessary and sufficient conditions that $\hat{A}$ is the zero matrix.

(b) State and prove necessary and sufficient conditions that $\tilde{A}$ is the zero matrix.