Linear algebra: how to evaluate a determinant

Tutoring college math, you cover determinants.  They are used in calculus, differential equations, and physics – just to name a few contexts.  However, they belong to linear algebra.  The tutor works a couple of examples.

The determinant is a number that arises from a matrix.  Consider the following example:

Example 1: Evaluate the determinant of matrx A:

Solution:

Multiply the numbers along the diagonal from top left to bottom right. Take that result, then subtract from it the product of the other diagonal:

det A = -2*-9 – 3*4 = 18-12=6

That’s fine – but what about a larger matrix? In fact, 3×3 matrices are probably the most common ones on which to evaluate the determinant. How do you do it, in that case?

Example 2: Find the determinant of matrix B:

Solution:

There are many ways to do this; here might be the most common:

Start at the top left. Imagine the square 2×2 matrix that results by omitting the first row and column (let’s call it matrix P). Multiply the top left number by det P:

3(2*5 – 0*(-7)) = 30

Continue with the top middle number in B: the 11. Now imagine the 2×2 matrix you get by omitting the top row and middle column. Do it the same as before, except you multiply it by -1 (the process flipflops between 1 and -1):

-1(11)(4*5 – (-1)(-7))= -11(13)=-143

Next step: repeat the process, this time from the top right number. The -1 from last step flip-flops back to 1. We proceed as follows:

1(4*0 – (-1)(2))=2

Finally, we take our three results from above and add them together:

30 – 143 + 2 = -111

So, the determinant of matrix B, above, is -111.

There is much to discuss about determinants. I’ll be saying much more about them in future posts:)

Source:

Johnson/Riess/Arnold: Introduction to Linear Algebra. Don Mills: Addison-Wesley, 1989.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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