College Algebra: Multiplying Matrices by a Scalar

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So we now know how to add and subtract matrices that are the exact same shape, that have the same dimensions. So that's not a big deal. Now in fact, let me just tell you a little more language. So these are examples of matrices. This is a 3 by 2, for example. This is a 3 by 1, and so on. Now the numbers al1 are just naked numbers by themselves, like a poor little 5 or a poor little 4. These are now referred to sometimes as scalars, just single elements, just a number rather than a whole list or an array of numbers. These are sometimes called scalars. So 1 thing we can do with matrices is to actually multiply them by scalars, and again, not a surprising outcome.
For example, if I take this matrix, which is a 3 by 1, and multiply it by 5, what do you think I would end up getting? Well I'd get another 3 by 1 matrix. And you'll never guess the answer. Yes, that's right. I take each of these elements and multiply it by 5. So I'd see, in this case, what would the answer be? Well this may just shock your world, but I would see 5, 15, -5, not a big deal. So scalar multiplication of a matrix by a scalar is just, literally, to take that scalar and multiply every single element by that scalar. So for example, this would give you another 3 by 2 matrix. And you would get 4, 16, 0, 28, -4 and 8. So again, it would be the exact same size, same shape. So scalar multiplication is not a big deal.
Just to illustrate that, in fact, we can combine these ideas together. I could take 2 people like this and ask for this operation. Let's take 5 times this, and add it to 4 times this. Put a plus sign here. So now I'm asking for matrix addition and scalar multiplication at the same time. This is getting tricky now. Well we can only add matrices that are the same shape. But these are the same shape, so we're okay there. And everything has to be multiplied by 5. And then I add it to what we get when we multiply this by 4. So in this place, I'd have a 5 and a 4. And when I combine them, I see a 9. Here I see a 15 and 8, and so that would be a 23. Here I see a 10 and a -4, so that's a 6. Here I see a 5 and a 0, so that's just 5. Here I see a 20 and a 12, so that's a 32. And here I see a 0 and a 20, so I get a 20. So again, scalar multiplication is not a big deal. You just literally multiply it through everywhere. And then you have another matrix of the same shape. Do the same thing here. And then you can combine or subtract or do whatever you want. So multiplication by a scalar, scalar multiplication, is not an issue.
Now multiplying 2 matrices together, you'll see.
Systems of Equations
Matrices
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