Now, matrix scalar multiplication, very similar idea.

Matrix addition, or matrix scalar multiplication, or the addition of two transformations.

And we also defined scalar times some matrix a to be equal to the scalar.

So if we want to multiply the scalar 5 times the matrix,

You can also multiply the whole matrix by a number, called a scalar.

If I were to think about the matrix kA, now I'm not just multiplying one row. I'm multiplying the whole matrix by a scalar.

A new matrix where each of its columns are the scalar times the column vectors of a.

You literally, if we go back to this definition, we can define scalar multiplication of a matrix.

So I just showed you that if I take my matrix and multiply it times some vector that was multiplied by a scalar first, that's equivalent to first multiplying the matrix times a vector and then multiplying by the scalar.

Well, this product is going to be each of these scalar terms times the column vectors of this matrix.

Well, once again, I haven't prove it to you yet, but it's actually a very straightforward thing to do, to show that when you're dealing with scalars, if you have a scalar here, it doesn't matter if you multiply the scalar times the vector before multiplying it times the matrix or multiplying the matrix times the vector, and then doing the scalar.

I'm not multiplying the whole matrix times the scalar. I can't say this is kA. I'm just multiplying one of its rows.

So the take away is the only way you can say that this is going to be some scalar multiple times your original determinant, is only if you multiply one row times that scalar multiple, not the whole matrix.

It's a scalar.

And, when you multiply a scalar, when you just multiply a number times the matrix, you just multiply that number times every one of its elements.

I did this because I wanted to show you also that multiplying a scalar times, or just a value or a number, times a matrix is just multiplying that number times all of the elements of that matrix.

They're just scalar numbers.

It's a scalar quantity.

This is just scalar multiplication.

This is a scalar quantitiy.

These are just scalar numbers.

This is the scalar multiple.

So we've just shown you in general, if you have any n by n matrix, if you multiply only one row, not the whole matrix, only one row by some scalar multiple k, the resulting determinant will be your original determinant times k.

This is the scalar x1 times the vector v1 plus the scalar x2 times the vector v2.

So now we have this definition that c times A, if I take any scalar times any matrix A, it's just equal to c times each of the column vectors.

This is just a scalar quantity.

That's a distance, that's a scalar.

So scalar multiplication has distributive properties.

I just multiplied out this scalar.

We've learned about matrix addition, matrix subtraction, matrix multiplication.

Five yards scalar, five yards deep vector.

We're multiplying by the scalar of x1.

We defined scalar multiplication times a transformation of x to be equal to a scalar times the transformation of x.

All of the possible scalar multiples of v are going to be the same thing as all of the scalar multiples of our unit vector, u, which is just a scalar multiple of v.

Our second condition is that the projection of a scalar multiple should be equal to a scalar multiple of the projection.

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

The scalar multiples, obviously, are any real number.

So we can rewrite A times x as being equal to the scalar x1 times the vector v1 plus the scalar x2.

This is a scalar quantity, this is a scalar quantity right over here. So if you want the acceleration due to gravity,

Sal, I already knew how to in algebra II in tenth grade or ninth grade, I already was exposed to multiplying a scalar times a matrix or adding two matrices with the same dimensions.

This matrix that I chose for an example is a good matrix. A non singular matrix. An invertible matrix.

So an identity matrix is a matrix.

Let's just do some matrix matrix products.

A scalar function called the electric potential can help.

Now, let's do the same thing with scalar multiplication.