[SOLVED] MATHS7027-Assignment 3

Description

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.

• 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.
• 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

• |AB|,
• |C⁻¹|,
• 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!)