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MATH 2568 PROBLEM SET 3

Due February 2, 2025

Problem 1 (Complete the proof of Theorem 3 in the notes).

Let A be an m × n matrix, C and D be n × p matrices, and ~v and w~ be n-vectors. As indicated in the notes,

(a) Use the identity (r+s)A = rA+sA for any scalars r and s to show that A(+) = A+A

(b) Use (a) to show that A(C + D) = AC + AD.

Problem 2. Express the vector  in the 2-dimensional plane R2 as the sum of a vector which lies on the line y = 3x and a vector on the line y = x/2.

Problem 3. Let A be an n ⇥ n matrix and ~v be the n-vector 

Suppose there exists a constant k such that the sum of the entries of any given column of A is k. Show that AT = k.

Problem 4. We’ve seen that matrix multiplication does not generally commute. However, it can happen that AB = BA for particular matrices A and B.

Find all matrices that commute with the matrix 

Problem 5.

(a) Find all 3 ⇥ 3 matrices A such that 

(b) Find all vectors ~v such that