We can also solve systems of linear equations using Cramer's Rule. It's pretty crazy. First we got to talk about a determinant.
Ok first I will say how to find a determinant, then we will talk about why we want to find a determinant. Remember, this can only be done for a square matrix.
So, here is how to find the determinant for $2 \times 2$ and $3 \times 3$ square matrices. Lets start with $2 \times 2$.
Nothing too crazy. Just multiply the top left value with the bottom right, and then add it to -1 times the top right and bottom left. I will say $3 \times 3$ matrices are a bit more complicated. There is the way I like to calculate them, the Sauss Method, and i think one more.
My way:
For each encircled colour at the top, multiply them by the determinant of the matrix made by the bottom four. There are 3 colours so that means this should be done 3 times. Add them all up, BUT SUBTRACT THE BLUE TERM. You should get an equation that looks like this:
There are a couple other ways to find a determinant of $3 \times 3$ matrices. One of them is Rule of Saurrus. A good representation is on Wikipedia, and I take it to put here:
Now if this was a course or something I'd stop right here and say "yep this is all u gotta know for the exam, you will be asked to calculate a 3x3 determinant on the homework, and a 2x2 on the exam." But, if you want to learn more, I will give you the knowledge you seek .
