Homework 1

This HW is due on 16th September 2024. Please write down the solutions neatly in your own handwriting, and submit the completed HW by email by the beginning of class on 16th September 2024. Please submit the HW as a single pdf file. To get credit, please provide all details and give complete reasoning for all your work.

Exercise 1 (2 points) Prove that any square matrix A can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

Exercise 2 (2 points) Prove that if a matrix is both symmetric and skew-symmetric, it must be a zero matrix.

Exercise 3 (2 points) Consider a n \times n matrix A. Show that if A is not invertible, then there is a n \times n matrix B with AB=0 but B \ne 0.

Exercise 4 (2 points) Let n < m. Consider a m \times n matrix A, and a n \times m matrix B. Prove that AB is not invertible.

Exercise 5 (2 points) Consider the 3 \times 3 matrix

A= \begin{bmatrix} 1 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \end{bmatrix}

Using the algorithm discussed in class (show all the steps), find (if it exists) the inverse of A^{-1}. (check your answer using the Python trinket on the webpage)