💪 Introduction to Matrix Algebra

Published

August 5, 2022

Directions

Complete the problems on this worksheet with your small group. You may want to refer to the Matrix Algebra for Educational Scientists text.

You will likely be learning (or re-encountering) many new mathematical terms. It is a good idea to note and define all the vocabulary/terms that you encounter as you work through this worksheet. You may want to do this individually or create a shared document that you can all contribute to.


Problems

Consider the following matrices:

\[ \mathbf{A} = \begin{bmatrix}3 & -2\\5 & 1\end{bmatrix} \quad \mathbf{B} = \begin{bmatrix}3 & -1\\-1 & 2\end{bmatrix}\quad \mathbf{C} = \begin{bmatrix}1 & 2 & 3\\0 & 1 & 2\end{bmatrix} \]

Make sure everyone in your group can solve each of these problem by hand and using R.

  1. What are the dimensions of A? C?

  2. Is C a square matrix? Explain.

  3. Find the trace of A.

  4. Find the determinant of A.

  5. Add A and B

  6. Find the transpose of C.

  7. By referring to the dimensions, can you compute AC? How about CA?

  8. Compute AC.

  9. Compute BI

  10. Create a \(3\times3\) diagonal matrix whose trace is 10.

  11. How do you know that B has an inverse? Explain.

  12. Compute \(\mathbf{B}^{-1}\)

  13. Create a \(3\times3\) matrix that has rank 2. Verify this using R.

  14. Create a \(3\times3\) matrix that is symmetric and is not I.

  15. Solve the system of linear equations using algebra (e.g., substitution, elimination) and then solve them using matrix methods (with R). To do this you will need to read the Systems of Equations chapter in Matrix Algebra for Educational Scientists.

\[ \begin{split} x + y + z &= 2 \\ 6x - 4y + 5z &= 31 \\ 5x + 2y + 2z &= 13 \end{split} \]