Lesson Determinants

Source code of 'Determinants'

This Lesson (Determinants) was created by by longjonsilver(2297)

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About longjonsilver: I have a new job in September, teaching

A determinant is a very power concept in linear algebra, as it helps the mathematician to solve equations quickly. I wont go into the theory of matrices here, as there are some good sites on the web already. I shall just offer a broken down version, that will hopefully get you started.

I will concentrate on a 3x3 matrix:

{{{
(
 matrix( 3, 3, 
  a[1], b[1], c[1],
  a[2], b[2], c[2],
  a[3], b[3], c[3]
 )
)
}}}

The determinant, D, for this is:

{{{D=a[1]*(b[2]*c[3]-b[3]*c[2]) - b[1]*(a[2]*c[3]-a[3]*c[2]) + c[1]*(a[2]*b[3]-a[3]*b[2])}}}

How do we "see" this?...well, there are 3 parts, each started with the top row, a1, b1 and c1. The signs of these cycle between + and -, so a 4x4 matrix would have a fourth term starting with -d1, for example.

What about the terms in the brackets? Well, consider the a1 term. Draw an imaginary vertical and horizontal line through the a1 term. The vertical line will run through the a2 and a3 terms. The horizontal line will run through the b1 and c1 terms. Effectively, we are left with the smaller 4-digit b2, c2, b3 and c3. We combine these as diagonal pairs, always top-left&bottom-right minus the bottom-left&top-right pair.

Why? Don't bother learning why at the moment, just learn the process.

Similarly, the second term: draw the imaginary vertical and horizontal lines to cross out a1 & c1 and the b2&b3 terms. This leaves the a2,a3 and c2, c3 terms, which are combined in exactly the same way as for the first term.

The third term follows the same process.

<b>and is this really useful? Too bloody right it is!</b>. See Cramer's Rule.