 # Section 6: Matrices

### Instructions

• This section covers the concepts listed below.
• For each concept, there is a conceptual video explaining it followed by videos working through examples.
• When you have finished the material below, you can go to the next section or return to the main Mathematical Probability page

### Concepts

Matrix Operations

### Exercises

Directions: There are no exercises for this topic.

### 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. Which matrices can be added?
2. Define matrix multiplication. Can we multiply any two matrices?
3. When are both $$AB$$ and $$BA$$ defined?
Gaussian Elimination

### Exercises

Directions: There are no exercises for this topic.

### 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. What are the three elementary row operations?
2. What is a row echelon form (REF)?
3. What is the objective of Gaussian elimination?
4. What is the reduced row echelon form of a matrix?
5. How do you use Gaussian elimination to see if a given matrix is invertible, and how does it help with finding the inverse?
Inverse of a Matrix

### Exercises

Directions: You should attempt to solve the problems first and then watch the video to see the solution.
1. Find the inverse of the matrix $A=\begin{pmatrix} 1 & -1 & -1 \\ 3 & -1 & 2\\ 2 & 2 & 3\end{pmatrix}$

$\mathbf{A}^{-1}= \begin{pmatrix} \frac{7}{10} & -\frac{1}{10} & \frac{3}{10}\\[8pt] \frac{1}{2} & -\frac{1}{2} & \frac{1}{2}\\[8pt] -\frac{4}{5} & \frac{2}{5} & -\frac{1}{5}\\[8pt] \end{pmatrix}$

### 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. Define the inverse of a matrix.
2. What is the significance of an invertible matrix?
3. What is the determinant of a matrix, and what is its significance in invertibility?