Question

Let $A$ be a $3\times 2$ matrix with real entries. Let $H=A(A^{T}A{)}^{-1}{A}^T$ where $A^T$ is the transpose of $A$ and let $I$ be the identity matrix of order $3\times 3$. Then

• A. $H^2=I$
• B. $H^2=-I$
• C. $H^2=H$
• D. $H^2=-H$

Answer 1

Answer: (C)

$H^2=H$

We know that from the propeties of inverse matrices,

${(A.B)}^{-1}=B^{-1}.A^{-1}$

$⟹H=A{A}^{-1}{\left(A^T\right)}^{-1}{A}^T=I⟹H.H=H^2=I.H=H$