If you solve the problem, you have probably something like this.

So let's do that.

It can set them down to 0.

So let's start with a 3 by 3 matrix and try to take the inverse.

Let me switch colors.

So the inverse of a is equal to 1 over the determinant.

So, if we carry out this calculation. And, again, I've spared you the details of inverting a matrix, 'cause it's actually a bit tedious.

Let me show you a quick example.

First let's put the data into octave.

So to solve for B, first, replace the predicted scores with the observed scores.

We figured out the determinant is negative 1 times the adjugate of a.

And it turns out there is very good numerical software for taking a matrix and computing its inverse.

And the matrix B is 3 I'm just going to pick random numbers minus 4, 2 minus 5.

5x5, you're almost definitely going to do a careless mistake if you did the inverse of a 5x5 matrix.

And as we'll see in the next video, calculating by the inverse of a 3x3 matrix is even more fun.

So, plugged it into R, checked it out, the hand calculations worked.

So by now hopefully you know how to add and subtract matrices as well as multiply them and you also know how, what are the definitions of the inverses and transposes of a matrix and these are the main operations used in linear algebra for this course.

We will now embark on what is probably my least favorite exercise or computation in mathematics and I think you'll see why where we will invert a 3 by 3 matrix.

And in my mind, the only thing less pleasant than inverting a 3 by 3 matrix is inverting a 4 by 4 matrix.

So if I have a matrix A, and that is a, b, c, d.

And, how about a second column?

And that's better left to a computer.

It is a three by three matrix.

It's A transpose times B times A that you subtract from the remaining omega prime, and the same for C over here.

So, in fact, in general, if you have a matrix operation like

Is that constant time, log time, or linear time? Same thing for an unordered list.

So how far did the column move?

I am going to pull out the second column of this, and do the corresponding calculation so there's this matrix times this vector 1, 2.

And as a concrete example.

You might have noticed that up here on the right, I made a very simple chart to try and explain how Fibonacci behaves to myself.

However, there are exceptions. Say I give you a bunch of the same type of transforms, such as four different translation matrices.

That way when we add 1, we get 1, which is the sides of the triangle.

But as you'll see, this is almost an exercise in not making careless mistakes.

In this video we'll talk about matrix addition and subtraction, as well as how to multiply a matrix by a number, also called Scalar Multiplication.

The mathematical notation for this is M superscript T, where T means transpose.

And if A is m by n then this identity matrix, right, for matrix multiplication make sense that has a m by n matrix because this m has a match up that m And in either case the outcome of this process is you get back to Matrix A, which is m by n.

A. So just be careful of that.

So you might see on an exam, figure out the determinant of A.

Finally, the last special matrix operation I want to tell you about is to do matrix transpose.

Perform arithmetic, scientific or financial calculations

It turns out that sometimes you can compute inverses by hand but almost no one does that these days.

So now let me ask you a question.

One thing you have to know for multiplication is for, in order to multiply two matrices they have to be what's called conformable.

But anyway, that is how you calculate the inverse of a 2x2.

Then here is my initialization of my probability table.