Question

If $X$ is any matrix of order $n\times p$ ($n$ and $p$ are integers) and $l$ is an identity matrix of order $n\times n$, then the matrix $M=I-X{\left(X^{\prime }X\right)}^{-1}{X}^{\prime }$ is
(i) Idempotent matrix
(ii) $MX=0$
Choose the correct answer

• A. (i) is correct
• B. (ii) is correct
• C. (i) is incorrect
• D. (ii) is incorrect

Answer 1

Answer: (A), (B)

(ii) is correct
(i) is correct

Given, $M=I-X{\left(X^{\prime }X\right)}^{-1}{X}^{\prime }$
$=I-X\left(X^{-1}{X^{\prime }}^{-1}\right)X^{\prime }$
$=I-\left(X{X}^{-1}\right)\left({X^{\prime }}^{-1}{X}^{\prime }\right)$
$=I-II=O$
$\therefore$ $M^2=M$ and $MX=O$