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).

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).