Home » mcq » Linear Algebra » A is m à n full rank matrix with m > n and $$I$$ is an identity matrix. Let matrix A’ = (ATA)-1AT, Then, which one of the following statement is TRUE? A. AA’ A = A B. (AA’)2 = A C. AA’A = $$I$$ D. AA’A = A’
AA' A = A
(AA')2 = A
AA'A = $$I$$
AA'A = A'
Answer is Wrong!
Answer is Right!
The correct answer is $\boxed{\text{C}}$.
Let $A$ be an $m\times n$ full rank matrix with $m>n$. Then, $A^T A$ is an invertible matrix. Therefore, $(A^T A)^{-1}$ exists.
We know that $A^T A A = A^T$. Therefore,
\begin{align}
AA’ 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z"/>
Subscribe on YouTube
A &= A (A^T A)^{-1} A^T \
&= A (A^T A)^{-1} A \
&= I.
\end{align}
Option A is not true because $AA’ A$ is an identity matrix, not $A$.
Option B is not true because $(AA’)^2 = AA’ A A’ = AA’$.
Option D is not true because $AA’ A$ is an identity matrix, not $A’$.