We are studying Linear Algebra topics in Precalculus class and this week’s lessons involved finding the inverse and determinant of a matrix. We need to find the inverse so we can solve X=(A-inverse)(B)
We used the formula in class to solve 2×2 matrices and also to calculate their determinants. Not that hard. Then we moved up to 3×3’s and then even tried a few 4×4’s.
When calculating the determinant, I reminded them to pick a row with the most zeros to reduce the amount of calculations but they are still a bit of challenge to complete. My students are getting very good at checking their work as they move along in the problem and documenting so they can double check the final determinant. When the determinant is zero the problem is done because there is no inverse to calculate.
For the inverse of a matrix, we followed standard procedure using Gaussian Elimination of the matrix on the left and the adjoining Identity matrix on the right. This method transforms the original matrix adjoined to the Identity matrix into the inverse of the original matrix. My students now have a great appreciation of the amount of calculations needed to find a 3×3 matrix inverse.
But there are many applications where the matrix is much larger, for example, ocean monitoring data can have much larger matrices. Also weather data collected from around the world and be analyzed in a matrix.
After completing the 3×3’s and a 4×4 matrix determinant and inversion, I asked the question, “How can we find the inverse of a larger matrix?”. We could use the calculators, but I thought a more realistic solution would be to use R in RStudio as this software package is free and I have experience using it for statistics. I generated a 10×10 and 11×11 matrix for the students to import and code in R to find both the determinant and inverse. I wrote some code examples to look at and I put a text file on Edmodo for them to import. I also showed them how to set up the matrix using the combine function.
The assignment is due next week as our computer lab was booked a few days this week. I also asked them to limit the amount of digits using the format function.
This topic has also led the class to wonder if there are other ways to find a matrix inversion. I read John Cook’s blog and he gives some advanced techniques on using Cholesky’s technique which I find very interesting and has me scrambling for my Linear Optimization textbook and other web links. This technique utilizes the upper-triangular and lower-triangular part of the matrix and this topic could be a good research paper for extra credit.
One last item that I find interesting is that I can use R and RStudio both at home on my Mac and at school on a PC. RStudio works exactly the same on both and it’s easy to start a lesson plan at home with an R script and finish it at school. This gives me time to adjust the lesson for each class. I’ll post next on the submitted work on Edmodo.
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