I can set up an augmented matrix where I put the identity matrix right there, just like that, and I perform a bunch of row operations.

The set of pivot columns for any reduced row echelon form matrix is linearly independent.

And actually, the columns of the identity matrix have a special I guess the set of columns has a special name.

Now, if we perform that same set of row operations on the identity matrix, what are we going to get?

So an identity matrix is a matrix.

Let's just do some matrix matrix products.

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.

In the last couple of videos, we've seen that if we have some matrix C that is n by n. It's a square matrix, and is columns, column form and orthonormal set.

A 2 by 3 matrix, a 2 by 3 times 3 by 2 matrix, I got a 2 by 2 matrix.

The identity matrix times any other matrix is just that matrix.

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So in general, the nullity of any matrix of any matrix

So it's a 2 by 3 matrix.

Anything times that identity matrix is going to be equal to itself.

It's actually going to be a 3 by 3 matrix, a much bigger matrix.

Cells Array

In mathematics, an integer matrix is a matrix whose entries are all integers.

This is equal to some new matrix I'll make it pretty big right here times our vector x .

Or, we would call this matrix, just by convention, you would call this a two by three matrix.

But now, since we assumed that these columns here form an orthonormal set, this just gets reduced to the identity matrix.

We have a 3 by 2, times a 2 by 3 matrix, so it's going to result in a 3 by 3 matrix.

I get the k by k identity matrix. I get the identity matrix of k, because it's going to be k by n, times n by k.

So that's my original A right there.

OK, so the matrix of cofactors is essentially applying this pattern to the matrix of minors.

Array of tools

I will create an augmented matrix.

It's a diagonal matrix.

Matrix of minors.

To figure out its transformation matrix, so if we say that T of x is equal to, I don't know, let's call it some matrix S times x.

If we have a 2x2 matrix, the identity matrix is 1, 0, 0, 1.

The transformation matrix with respect to that eigenbasis, it becomes a very, very nice matrix.

We multiplied the 3 by 2 matrix times a 2 by 3 matrix, and we got a 3 by 3 matrix.

I could just create a coefficient matrix, where the coefficient matrix would just be, let me write it neatly, the coefficient matrix would just be the coefficients on the left hand side of these linear equations.

Well our change of basis matrix is just the matrix with these guys as the columns.

We have a 3 by 4 this right here is a 3 by 4 matrix, and this guy right here is a member of R2.

So the reason why I'm going all this is because we just did a matrix matrix product up here.

This times this will equal the identity matrix.

And it turns out there is such a matrix.

So it's an m by n matrix.

So, let me draw a matrix for you.

That is a five by two matrix.

This is a 3 by 2 matrix.

That's a 1 by 2 matrix.

Let's say that I have a list

And this is going to be the identity matrix.