Following up on yesterday’s post, the tutor continues about determinants. Tutoring university math or natural sciences, they come up often.
Yesterday’s post covered some basics about determinants including a 2×2 and a 3×3 example. Although it revealed the necessities for many practical situations, it left out most of the theory. However, determinants are a playground for mathematicians; covering all the theory about them in a dozen posts would still be impossible.
We face the following question: are there little bits of extra theory that could really help a student with determinants? Are there little observations that could ease a student’s uptake of the topic?
Today: one observation and one bit of theory:
Observation 1: When a matrix is shown in vertical brackets rather than square ones, it usually means the determinant of the matrix.
That is, if you have matrix A:
Theoretical point 1:
You can expand the determinant along any row or column. In yesterday’s example, I showed how to evaluate the determinant of B
from the top row. However, you could evaluate det B from the middle column instead. For the negative flip-flopping, remember to multiply each step by (-1)^(r+c), where r is the row, and c is the column.
I’ll now evaluate det B from the middle column (using the procedure from my previous post):
The middle column starts at 11, which is in row 1, column 2. Therefore, the “flip-flop” factor will be (-1)^(1+2)=-1. Imagining the matrix without the first row and second column, we proceed:
We move to the next number in the second column: the 2. Its flip-flop factor is (-1)^(2+2)=1:
1×2(3*5 – (-1)(1))=2(16)=32
Now, we arrive at the third member of the middle column, which is a 0. Here we get a break: 0 times anything else is 0.
Finally, we add our results:
So, det B comes to -111, just as it did from yesterday’s expansion along the top row.
Having the freedom to choose the row or column to expand from is definitely an advantage when evaluating the determinant, since you can make convenient use of zeros in a matrix.
I hope this helps all you college/university students, for whom first term exams draw near:)
Source: Johnson|Riess|Arnold. Introduction to Linear Algebra, 2nd Edition. Don Mills, Ontario: 1989.
Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.