(Solved): 1. (Exercise 1) Let \( A \) be an \( n \times n \) matrix and \( I \) the \( n \times n \) identit ...

1. (Exercise 1) Let \( A \) be an \( n \times n \) matrix and \( I \) the \( n \times n \) identity matrix. Assume that \( A \) is invertible (nonsingular). Perform the following multiplications: (a) \( A^{-1} \cdot[A \mid I] \). (b) \( [A \mid I]^{t} \cdot[A \mid I] \), where \( B^{t} \) denotes the transpose of \( B \). (c) \( [A \mid I] \cdot[A \mid I]^{t} \). (d) \( \left[\frac{A^{-1}}{I}\right] \cdot[A \mid I] \). (e) \( [A \mid I] \cdot[A \mid I]^{t} \). (f) \( \left[\frac{A}{I}\right] \cdot A^{-1} \).