So an identity matrix is a matrix.

Let's just do some matrix matrix products.

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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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 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.

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So turn this from a vector, vector dot product to a matrix, matrix product.

So to convert from a matrix of minors to a matrix of cofactors, you just have to remember this pattern.

That would get me that much closer to the identity matrix.

And where did that 3 by 3 come from?

So if we have one matrix A, and it's an m by n matrix, and then we have some other matrix B, let's say that's an n by k 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.

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.

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

When you take a matrix and you multiply it times each of the column vectors, when you transform each of the column vectors by this matrix, this is the definition of a matrix matrix product.

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.

So I'm a little bit closer to having the identity matrix here.

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.