I had a difficult time making sense of this topic. Good mathematicians frequently have trouble teaching math—I’m guessing—because they have mastered the art of thinking in numbers but are unable to translate that into English. Or perhaps—I’m also guessing—a subject like determinants are such a masterpiece of numerical thinking that it compares to a speech from one of Shakespeare’s plays. Nonetheless, to the rest of us who think primarily in English, it goes right over our heads.
I could not get a sensible answer to these simple questions: Why is calculating the Determinants important or significant? How are they useful? Michael Kelley in his Algebra, pg 120 tells us “…at the moment, don’t focus on why you find determinants, focus on how you find them…” He promises to show us how to use determinants with Cramer’s Rule. But M. Bourne on the Intmath.com site demonstrates with the equation below that when you calculate the determinants, you don’t need to use Cramer’s Rule. No less authority than Lancelot Hogben, in his Mathematics for the Million writes that determinants were first developed by the Chinese, perhaps for solving magic puzzles, later they were developed in the western world by Chess players. Hogben says they are a labor saving way of solving simultaneous equations. But then, Hogben too gets side tracked into a lengthy explanation of how to calculate them that ironically looks worse than just solving the equations by elimination of variables. (See Chapter 8, Kelley Algebra.) The article on determinants in Wikipedia is filled with too much incomprehensible jargon.
What to do? The non-mathematical mind rebels at pointless drudgery.
Fortunately, I had an epiphany when I looked at the answer to the Determinant Exercise 2 on the intmath.com site. Once they had calculated the determinant then they used that determinant to solve the equation through division. (With no need for Cramer’s Rule.) There was my answer. That is how the J command (verb) Matrix Divide (%.) solves the equations through division of the matrix.
In Determinant Exercise 2 on the intmath.com site, there is a system of equations to solve that looks much like Gary Helzer’s $3.00 I-APL set of equations. Here are the equations:
x + 3y + z = 4
2x – 6y – 3z = 10
4x – 9y + 3z = 4
So, I set up these equations in J in the same way as Helzer’s equations:
You have to click the Answer button under Exercise 2 on the intmath.com site to get the Answer and the explanation:
4 2/3 _2
2/3 is equal to .666667. In explaining the steps in their Exercise 2, Matrix A is what they used to calculate the determinant for this equation. The answer they got is -93. Then they divided all of the variables by the determinant, -93, to find the values of x, y. They solved for z by elimination. Using J to find the determinant, I got the same answer -93 as shown above.
So, after looking at the pencil and paper calculations in the answer to Exercise 2 on the Intmath.com site , I think J Matrix Divide automatically calculates the determinant, and then finds the value of x, y, and z by division using the determinant. In both this equation and Helzer’s equation in J, using Cramer’s rule is unnecessary. So that is specifically how the determinant is used. In J, Matrix Divide is a genuine labor saving method. Next post will show how to calculate 2×2 determinants.