MATHS 7027 Mathematical Foundations of
Data Science
Assignment 3
(PDF only)
1. Consider the matrix
,
and the general 2 × 2 matrix
.
Find the conditions on a, b, c, and d, such that A and B commute (i.e., AB = BA). Therefore, write out the most general form of the matrix B that commutes with A.
2. Consider the system of equations
x1 + 4x2 − 6x3 − 3x4 = 3
x1 − x2 + 2x4 = −5 x1 + x3 + x4 = 1 x2 + x3 + x4 = 0.
(a) Convert this system to augmented matrix form and solve using Gauss-Jordan elimination (or explain why no solution exists). Make sure you show all of your steps, by writing out the row operations performed on the augmented matrix.
(b) Check your answer either by hand by performing an appropriate matrix multiplication on your solution (x1,x2,x3,x4).
3. Consider the homogeneous system of equations
x + y + z = 0
2x − 6y − 2z = 0
2x + z = 0.
Convert to augmented matrix form and use Gauss-Jordan elimination to find all solutions of the system of equations (i.e., not just the trivial solution x = y = z = 0).
4. Consider the matrices
.
Find:
1
(a) |AB|,
(b) |C−1|,
(c) The set of all values of x such that the matrix D is invertible. (Note: make sure that you use appropriate set notation to write your answer!)
2
