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

So let's do that.

It can set them down to 0.

Let me switch colors.

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

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.

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

Let me show you a quick example.

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

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

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

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.

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

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.

First let's put the data into octave.

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

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

And that's better left to a computer.

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

And as a concrete example.

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

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

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

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.

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 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.

And, how about a second column?

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.

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

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

It is a three by three matrix.

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

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

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

So now let me ask you a question.

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

Perform arithmetic, scientific or financial calculations

So how far did the column move?

Since we want to test the two halves of the input separately, we need to split it into two parts.

Then here is my initialization of my probability table.

Matrix multiplication is really useful since you can pack a lot of computation into just one matrix multiplication operation.

Then we don't have anything to do with the 3.