 # Section 10: Fundamental Matrices

### Instructions

• First, you should watch the concepts videos below explaining the topics in the section.
• There are no exercises in this section, but you can see the exercises in Sections 8, 9, and 11, which include finding the fundamental matrix.
• Last, you should attempt to answer the self-assessment questions to determine how well you learned the material.
• When you have finished the material below, you can start on the next section or return to the main differential equations page.

### Concepts

• Definition of a fundamental matrix for a system of equations
• Properties of a fundamental matrix

### Exercises

Directions: See the exercises in Sections 8, 9, and 11. Those videos all solve system of equations, and then show how to find the fundamental matrix for the system.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. How is a fundamental matrix of a linear homogeneous system of ODEs defined? Is it an invertible matrix? What is the determinant called?
2. How do you write the general solution of the system if you have a fundamental matrix for it?
3. What is the special fundamental matrix of an IVP system of ODEs set up at $$t=t_0$$? What is its value at $$t_0$$?