1. **Problem Statement:** Determine if the statement "If $A$ is an $n \times n$ matrix with $n$ linearly independent columns, then $A^2$ is invertible" is true or false. 2. **Recall the definitions and properties:** - A matrix $A$ with $n$ linearly independent columns is invertible. - If $A$ is invertible, then there exists $A^{-1}$ such that $A A^{-1} = I$, where $I$ is the identity matrix. - The product of invertible matrices is invertible, and the inverse of the product is the product of the inverses in reverse order. 3. **Apply these properties to $A^2$:** Since $A$ is invertible, $A^2 = A \times A$ is a product of two invertible matrices. 4. **Conclusion:** Therefore, $A^2$ is invertible, and its inverse is given by $$ (A^2)^{-1} = A^{-1} A^{-1} = (A^{-1})^2. $$ **Final answer:** The statement is **true**.