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
asked 2021-02-21
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?

Answers (1)

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
So the answer is false.
The correct option is (2).
0

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