# Let A,B be two matrices with linearly independent columns . If Col(A)=Col(B) then A(A^TA)^{-1}A^T=B(B^TB)^{-1}B^T True or False?

Let A,B be two matrices with linearly independent columns . If $Col\left(A\right)=Col\left(B\right)$ then $A\left({A}^{T}A{\right)}^{-1}{A}^{T}=B\left({B}^{T}B{\right)}^{-1}{B}^{T}$
True or False?
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Step 1
Let A and B are two matrices.
And A and B are linearly independent columns.
Step 2
If col(A) =col(B)
Since A and B are linearly independent.
So $\alpha A+\beta B=0$
So, the matrices are orthonormal,
So $A{A}^{T}=B{B}^{T}$
From this,
$\left(A{A}^{T}{\right)}^{-1}=\left(B{B}^{T}{\right)}^{-1}$
Then,
$A\left({A}^{T}A{\right)}^{-1}{A}^{T}=B\left({B}^{T}B{\right)}^{-1}{B}^{T}$
Since in matrices,
$A{A}^{T}=\ne {A}^{T}A$
The correct option is (2).

Jeffrey Jordon