1. Students will apply step-by-step row operations to a matrix to decompose it into the product of two matrices: a lower triangular matrix and an upper triangular matrix (LU decomposition).
2. Students will use basic MATLAB commands to implement the sequence of operations leading to the matrix decomposition. They will also use MATLAB to verify the correctness of their resulting decomposition.
3. Students will establish connections between linear algebra theory and the design of computational methods in numerical analysis.
4. Students will practice digital documentation by presenting their computations, results, and verification.
5. Through the project, students will engage in higher level thinking by integrating mathematical representation of ideas, computational implementation, and proper documentation of their findings.

Students need a solid background in linear algebra before learning the topic of matrix decomposition.

The activity requires a MATLAB license.

This activity consists of three parts that guide students from learning basic MATLAB commands to implementing and documenting a matrix decomposition.

Part A: Introduction to MATLAB and Row Reduction
Part A serves as an introduction on using MATLAB commands in linear algebra. Students learn how to enter matrices into MATLAB, perform elementary row operations, and obtain row echelon form and reduced row echelon form of a matrix.

Students complete an assignment in which they use the MATLAB commands to carry out the operations. They need to show both commands and outcome of their implementations.

(1) Input the matrix A = [2, 4, 3, 5; 1, 2, -1, 4; 1, 2, -6, 7] into MATLAB.
(2) First Approach: Perform a sequence of elementary row operations step by step in MATLAB to obtain the reduced row echelon form (R-REF) of the matrix.
(3) Second Approach: Use MATLAB's built-in rref command to compute the reduced row echelon form directly.
(4) Compare the results obtained from the above two approaches and see whether they are the same.

Part B: LU Decomposition with Permutations
Part B introduces LU decomposition where row permutations might be required in the process. This part emphasizes the relationship between elimination operations, permutation matrices, and matrix factorization.

(1) In MATLAB, first define a 4 by 4 matrix A = [1, 1, 1, 0; 1, 1, -1, 0; 1, -1, 0, 1; 1, -1, 0, -1].
(2) Identify a sequence of permutation matrices P1, P2, P3 and elimination matrices E1, E2, E3 to transform A into an upper triangular matrix U. That is, E3 P3 E2 P2 E1 P1 A = U.
(3) Show that E3 P3 E2 P2 E1 P1 A = U can be written as E3' E2' E1' P3 P2 P1 A = U, what would be the matrices E1', E2', and E3'? 
(4) Based on your work in the previous step, determine the matrices P, L and U for the given matrix A such that PA = LU, where L is a lower triangular matrix and U is an upper triangular matrix.
(5) Verify in MATLAB that PA=LU using the P, L, and U you found in step (4).

Part C: Documentation, Presentation, Reflection, and Revision

Students prepare a digital report that shows the implementation process, theoretical reasoning, and the verification of results. They will have an opportunity to present their work and answer questions from peers and the instructor for further revision and improvement.

The activity can be introduced and specified in one lecture. Students will have one week to complete Parts A and B. The lecture provides an opportunity for students to ask questions and know expectations. The instructor offers hints and guidance to support successful completion of the assignment.

Parts A and B are manually graded, with points awarded for each answer based on its correctness and completeness.
Part C is intended for further improvement, and it provides a channel for the instructor to give individualized feedback to students, but no additional points will be awarded. Nevertheless, all students can benefit from the feedback provided.