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The question asks about the least squares solution, focusing on the matrix \( A \) where \( \text{rank}(A) = \text{number of columns of } A \). This implies that \( A \) has full column rank, meaning \( A^T A \) is invertible.

In least squares problems, the solution to \( A x = b \) in the over-determined case (more equations than unknowns) is given by \( x = (A^T A)^{-1} A^T b \). This formula represents the unique least squares solution because \( A^T A \) is invertible.

### Correct Answer: 
The second option:
\[
A(A^T A)^{-1} A^T b \text{ is the unique least square solution to } Ax = b.
\]
is **always true** in this case. 

Would you like a detailed explanation or have any specific questions? Here are five related questions to consider:

1. What does it mean for a matrix to have full column rank?
2. Why is \( A^T A \) invertible when \( \text{rank}(A) = \text{number of columns of } A \)?
3. What is the difference between a least squares solution and an exact solution?
4. How is the least squares solution derived from normal equations?
5. What happens to the least squares solution if \( A^T A \) is not invertible?

**Tip**: In least squares problems, the projection \( A u = b \) implies that the solution minimizes the error \( \| A x - b \| \).

Suppose rank(A) = number of columns of A. Which of the statements is always true?

Learn about the unique least squares solution when a matrix A has full column rank, focusing on the formula (A^T A)^{-1} A^T b and its application in solving Ax = b. Ideal for university-level students studying linear algebra.

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