# 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?

Question
Matrices
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?

2021-02-22
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 $$\alphaA +\betaB =0$$
So, the matrices are orthonormal,
So $$AA^T =BB^T$$
From this,
$$(AA^T)^{-1} =(BB^T)^{-1}$$
Then,
$$A(A^TA)^{-1}A^T=B(B^TB)^{-1}B^T$$
Since in matrices,
AA^T= \neq A^TA
The correct option is (2).

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