81. Let A be any n x n symmetric idempotent matrix and let B be an (m

81. Let A₁₁ be any n x n symmetric idempotent matrix and let B be an (m + n)

x (m + n) matrix defined by

$$B =

\begin{bmatrix}

A_{11} & A_{12} \\

A_{21} & A_{22}

\end{bmatrix}.$$

Show that B is a symmetric idempotent matrix if and only if A₂₂ is a symmetric idempotent matrix and A₁₂ = 0.