But of course, if I multiplied the inverse matrix times the identity matrix, I'll get the inverse matrix.

Calculates the inverse of the matrix.

What's the inverse of that matrix?

So this guy right here, this series of matrix products, this must be the same thing as my inverse matrix, as my inverse transformation matrix.

We've actually managed to inverse this matrix.

And a square matrix for which there is no inverse, of which an inverse is undefined is called a singular matrix.

Perhaps even more interesting than finding the inverse of a matrix is trying to determine when an inverse of a matrix doesn't exist.

Well this is the inverse of this original matrix.

So what if we knew the matrix a inverse?

And so what's the inverse of the identity matrix?

So A transpose A inverse becomes the inverse of the k by k identity matrix, which is just the k by k identity matrix.

The matrix multiplied with its inverse results in the unity matrix of the same dimension.

And we wanted to find the inverse of this matrix.

The transpose of this matrix is equal to the inverse.

Now, in the last video we said that the inverse matrix, so if this is Tnaught, Tnaught inverse could be represented it's also a linear transformation It can be represented by some inverse matrix that we just called A inverse times x.

So the other way to view that is, if a determinant of any matrix is equal to 0, then that matrix is a singular matrix, and it has no inverse, or the inverse is undefined.

It's called Gauss Jordan elimination, to find the inverse of the matrix.

If we call this the matrix a, let's figure out a inverse.

That's the identity matrix, times x is equal to a inverse b.

What I want to do is use our technique for finding an inverse of this matrix to essentially find a formula for the inverse of a 2 by 2 matrix.

We've already seen that the transpose of this matrix is the same thing as the inverse of this matrix.

So I multiplied this by a inverse, to get to the identity matrix.

So A times A inverse should also be equal to the identity matrix.

But let's confirm that this really is the inverse of the matrix B.

So that's just the matrix x, or the vector x times a inverse b.

So we have a generalized way of figuring out the inverse for transformation matrix.

And we saw that the inverse transformation matrix times our transformation matrix is equal to the identity matrix. We saw this last time. We proved this to you.

That means that there's some C inverse that exists, such that C inverse times C is equal to the n by n identity matrix.

If these matrices are collectively the inverse matrix, if I do them, if I multiply the identity matrix times them the elimination matrix, this one times that equals that.

So the inverse of matrix a is equal to 1 over the determinant of a times the adjugate, or adjoint, of matrix a.

I'm essentially multiplying when you combine all of these a inverse times the identity matrix.

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

The composition of the inverse with the function has to become the identity matrix on x.

So this matrix is singular, or it has no inverse, if this expression is equal to 0.

And then we have C inverse, but because C was a square matrix with orthonormal columns, we know that C inverse is the same thing is C transpose.

And this right here, this is C inverse, which is equal to C transpose, because C is an invertible matrix, or a square matrix, with orthonormal columns.

I will now show you my preferred way of finding an inverse of a 3 by 3 matrix.

It's saying that the composition of f inverse with f, has to be equal to the identity matrix.

In the last video, we stumbled upon a way to figure out the inverse for an invertible matrix.

So you apply those same transformations to the identity matrix, you're going to get the inverse of A.

That if we have our change of basis matrix, we have it's inverse, and we have just our standard basis linear transformation matrix, we're able to get this.

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

I multiply it by this matrix and I'll call that the inverse of a is there a matrix where I'm left with, not the number 1, but I'm left with the 1 equivalent in the matrix world?

And then when I have the identity matrix on the left hand side, what I have left on the right hand side will be the inverse of this original matrix.

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