Linear Algebra – Prove Matrices Have Equal Rank

linear algebramatrices

If $P$ and $Q$ are $n \times n$ matrices of real numbers such that

  1. $P^2=P$ and
  2. $Q^2=Q$ and
  3. $I-P-Q$ is invertible where $I$ is an $n \times n$ identity matrix,

Show that $P$ and $Q$ have the same rank.

If $P$ is non-singular, then it can be shown that $P=Q=I$, so they have same rank. But I can't prove it when $P$ is singular.

Best Answer

Hint. If $A$ is an invertible matrix, then $\operatorname{rank}(X)=\operatorname{rank}(XA)$ and $\operatorname{rank}(Y)=\operatorname{rank}(AY)$ for all square matrices $X$ and $Y$.

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