Lab 2. Matrix Algebra: Interactive Practice

This lab accompanies Chapter 2: Matrices as Machines: Algebra, Inverses, Rank, and LU.

The goal is to connect several views of matrix algebra:

Algebra: matrix products, inverses, transposes, and powers.
Geometry: matrices as transformations of the plane.
Computation: rank, determinant, Gram matrices, and LU factorization.

Python practice notebook

You may also use the Jupyter notebook version for longer Python practice:

More examples:

Interactive lab

Submission questions

Submit short written answers to the following questions.

Question 1. Matrix multiplication and order

Use the interactive lab to find two $2\times 2$ matrices $A$ and $B$ such that

\[ AB\neq BA. \]

Record:

$A$;
$B$;
$AB$;
$BA$.

Explain why the order of matrix multiplication matters when matrices are interpreted as transformations.

Question 2. Invertibility and information loss

Create one invertible matrix $A$ and one singular matrix $S$.

For each matrix, record:

determinant;
rank;
whether the inverse exists;
the geometric effect on the unit square.

Explain why singular matrices lose information.

Question 3. Transpose and Gram matrix

For one matrix $A$, compute

\[ A^T,\qquad A^TA. \]

Explain why $A^TA$ is symmetric. What information does $A^TA$ record about the columns of $A$?

Question 4. LU factorization

Use the LU tool on the Chapter 2 example

\[ A= \begin{bmatrix} 2&1&1\\ 4&-6&0\\ -2&7&2 \end{bmatrix}. \]

Record $L$ and $U$, and verify that

\[ A=LU. \]

Question 5. Pivoting

Use the LU tool on a matrix that needs pivoting. Explain why the ordinary LU algorithm fails without row swaps.

Question 6. AI companion

Ask an AI tool:

Explain why $A(Bx)$ is often faster to compute than $(AB)x$.

Then critique the response.

Your critique should answer:

Does the response compare the number of operations?
Does it mention matrix-vector multiplication versus matrix-matrix multiplication?
Does it correctly say that the two expressions are mathematically equal when dimensions match?
Does it explain why implementation choices matter?

# Lab 2. Matrix Algebra: Interactive Practice {.unnumbered} This lab accompanies **Chapter 2: Matrices as Machines: Algebra, Inverses, Rank, and LU**. The goal is to connect several views of matrix algebra: 1. **Algebra:** matrix products, inverses, transposes, and powers. 2. **Geometry:** matrices as transformations of the plane. 3. **Computation:** rank, determinant, Gram matrices, and LU factorization. ## Python practice notebook You may also use the Jupyter notebook version for longer Python practice: - [Download Lab 2 notebook](lab-02-matrix-algebra.ipynb) - [Open Lab 2 in Google Colab](https://colab.research.google.com/github/wanghemath/MATH5110/blob/main/Labs/lab-02-matrix-algebra.ipynb) More examples: - [Chromaticity](https://colab.research.google.com/github/wanghemath/MATH5110/blob/main/Labs/Lab_2a_Chromaticity.ipynb) - [Circular Orbit](https://colab.research.google.com/github/wanghemath/MATH5110/blob/main/Labs/Lab_2b_Circular_Orbit.ipynb) - [Determinant](https://colab.research.google.com/github/wanghemath/MATH5110/blob/main/Labs/Lab_2c_Areas_Determinants.ipynb) ## Interactive lab ```{=html} <iframe src="lab-02-interactive.html" width="100%" height="1850" style="border:1px solid #ddd; border-radius:12px;"> </iframe> ``` ## Submission questions Submit short written answers to the following questions. ### Question 1. Matrix multiplication and order Use the interactive lab to find two $2\times 2$ matrices $A$ and $B$ such that $$ AB\neq BA. $$ Record: - $A$; - $B$; - $AB$; - $BA$. Explain why the order of matrix multiplication matters when matrices are interpreted as transformations. ### Question 2. Invertibility and information loss Create one invertible matrix $A$ and one singular matrix $S$. For each matrix, record: - determinant; - rank; - whether the inverse exists; - the geometric effect on the unit square. Explain why singular matrices lose information. ### Question 3. Transpose and Gram matrix For one matrix $A$, compute $$ A^T,\qquad A^TA. $$ Explain why $A^TA$ is symmetric. What information does $A^TA$ record about the columns of $A$? ### Question 4. LU factorization Use the LU tool on the Chapter 2 example $$ A= \begin{bmatrix} 2&1&1\\ 4&-6&0\\ -2&7&2 \end{bmatrix}. $$ Record $L$ and $U$, and verify that $$ A=LU. $$ ### Question 5. Pivoting Use the LU tool on a matrix that needs pivoting. Explain why the ordinary LU algorithm fails without row swaps. ### Question 6. AI companion Ask an AI tool: > Explain why $A(Bx)$ is often faster to compute than $(AB)x$. Then critique the response. Your critique should answer: 1. Does the response compare the number of operations? 2. Does it mention matrix-vector multiplication versus matrix-matrix multiplication? 3. Does it correctly say that the two expressions are mathematically equal when dimensions match? 4. Does it explain why implementation choices matter?