Finding the determinant

To find the determinant of a matrix, there are 2 different methods that can be used, depending on the size of the matrix.

In a 2x2 matrix (two columns, two rows) the determinent is found by:

1. multiplying the two terms on the main diagonal
2. multiplying the two terms on the other diagonal
3. subtracting the result in (2) from (1), ie. (1) - (2) = determinant
   

In other sized matricies, the determinent is found in a few more steps:

1. write down the term in the top-left corner. Now, draw a line through all terms below it and in line with it. This leaves a smaller matrix. Write this matrix in brackets, next to the term you wrote earlier. You will multiply this later.
2. Repeat step 1 for all the terms across the top row. Note you must follow the 'checkerboard of signs' when doing this.
3. Now with all the terms you've got written down, find the determinents of the smaller matricies and multiply them by the term before it.
4. Once you have done this, you have the determinant (click on the image to enlarge and see a graphic representation of this)
   

DEFINITIONS (in basic terms)
Checkerboard of signs - when you need to use this, it means that every second term must be multiplied by (-1) in the first row.
iff - used commonly in mathematics. Means if and only if.
Main Diagonal - the diagonal that runs from the top-left to the bottom-right of the matrix

EXAMPLE

An example can be found in the image on the right. If you cannot read it, click on it to enlarge.


IS A MATRIX SINGULAR?
A matrix is singular if and only if (iff) the determinant of that matrix is zero.

Homogenous linear differential equations

To find the 'general solution' of equations that look like these:

y'' + 2y' + 3y = 0
2y'' - y' - 6y = 0
y'' - 3y' - y = 0

Simply follow these steps:

1) Rewrite the equation in terms of 'm', and for each (') count as a power (ie. ' -> x¹)

2) Now find the roots (or find a value for m). You can do this by either:
a) splitting the middle term then factorising, or
b) use the quadratic equation


3) Now you must 'sub' those roots into one of the three 'general solutions.' Which equation you must use is dependant on what your roots (m) are.

If roots are REAL and DISTINCT: use y = c1 e^mt + c2 e^mt
If roots are REAL but THE SAME: use y = c1 e^mt + c2 (t) e^mt
If roots have a REAL part and an IMAGINARY part: use y = e^rt(c1cosqt+c2sinqt)

Then you have the general solution.

EXAMPLE

Find the general solution of y'' + 2y' - 3y = 0

1) m² + 2m - 3 = 0

2) You can use either method (a) or (b) to find the roots. Both are shown:

a) a=1, b=2, c=-3
The two numbers 3 and -1 will add to b and multiply to c
m² - m + 3m - 3 = 0
m(m-1)+3(m-1) = 0
(m+3)(m-1) = 0
m=-3, 1

b) Using the quadratic equation, the outcome is m=-3 or m=1

3) Roots (m) are REAL and DISTINCT.
So... use y=c1 e^mt + c2 e^mt (see graphic above)

Sub in m=-3 and 1: y = c1 e^-3t + c2 e<sup>t

That is the general solution!</sup>