How do matrices multiply

Thus, the second entry will be. Now we are going to use the same strategy to look for the last two entries. This gives us:. Now we are done! This is what we get when we are multiplying 2 x 2 matrices. In general, the matrix multiplication formula for 2 x 2 matrices is. Now the process of a 3 x 3 matrix multiplication is very similar to the process of a 2 x 2 matrix multiplication.

Again, why don't we do a matrix multiplication example? If we are to keep locating all the entries and doing the dot product corresponding to the rows and columns, then we get the final result. We are done! Notice that the bigger the matrices are, the more tedious matrix multiplication becomes.

This is because we have to deal with more and more numbers! In general, the matrix multiplication formula for 3 x 3 matrices is. So far we have multiplied matrices with the same dimensions. In addition, we know that multiplying two matrices with the same dimension gives a matrix of the same dimensions.

But what happens if we multiply a matrix with different dimensions? How would we know the dimensions of the computed matrix? First, we need to see multiplying the matrices gives you a defined matrix.

There are cases where it is not possible to multiply two matrices together. For those cases, we call the matrix to be undefined. How can we tell if they are undefined? The product of two matrices is only defined if the number of columns in the first matrix is equal to the number of rows of the second matrix.

First, notice that the first matrix has 3 columns. Also, the second matrix has 3 rows. See that the first number is 2 and the last number is 4. Now that we know the dimensions of the matrix, we can just compute each entry by using the dot products. This will give us:. Now that we know how to multiply matrices very well, why don't take a look at some matrix multiplication rules? So what type of properties does matrix multiplication actually have?

First, let's formally define everything. If all five of these matrices have equal dimensions, then we will have the following matrix to matrix multiplication properties :.

The associative property states that the order in which you multiply does not matter. See how the left side and right side of the equation are both equal.

Hence, we know that the associative property actually works! Again, this means that matrix multiplication order does not matter! Now the next property is the distributive property. The distributive property states that:. We see that we are allowed to use the foil technique for matrices as well. Just to show that this property works, let's do an example. Hence computing that gives us:. Now let's check if the right hand side of the equation gives us the exact same thing.

Computing this gives us:. Notice that the left hand side of the equation is exactly the same as the right hand side of the equation. Hence, we can confirm that the distributive property actually works. We know that matrix multiplication satisfies both associative and distributive properties, however we did not talk about the commutative property at all.

Does that mean matrix multiplication does not satisfy it? It actually does not, and we can check it with an example. Question 8 : If matrix multiplication is commutative, then the following must be true:. First we compute the left hand side of the equation.

Now there are still a few more properties of the multiplication of matrices. However, these properties deal with the zero and identity matrices. The matrix multiplication property for the zero matrix states the following:. This is means that if you were to multiply a zero matrix with another non-zero matrix, then you will get a zero matrix. Let's test if this is true with an example. Now what about the matrix multiplication property for identity matrices? Well, the property states the following:.

Again, we can see that the following equations do hold with an example. So the equation does hold. Again, the equation holds. So we are done with the question, and both equations hold. This concludes all the properties of matrix multiplication. Now if you want to look at a real life application of matrix multiplication , then I recommend you look at this article. Solving a linear system with matrices using Gaussian elimination. Back to Course Index.

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Intro Lesson. Lesson: 1a. Lesson: 1b. Lesson: 1c. Lesson: 1d. Lesson: 2a. Lesson: 2b. Lesson: 2c. Lesson: 3a. Lesson: 3b. Lesson: 3c.

Lesson: 3d. Intro Learn Practice. Matrix Multiplication There are exactly two ways of multiplying matrices. Scalar Multiplication scalar multiplication is actually a very simple matrix operation. Equation 1: Scalar Multiplication Example 1 pt.

Equation 2: Scalar Multiplication Example 2 pt. Equation 3: Dot Product Example pt. Equation 4: Dot Product Failure Example pt. Equation 5: 2 x 2 Matrix Multiplication Example pt.

Just so I remember what I'm doing, let me copy and paste this. And then I'm going to get out my little scratch pad. So let me paste that over here. So we have all the information we needed. And so let's try to work this out. So matrix E times matrix D, which is equal to-- matrix E is all of this business. So it is 0, 3, 5, 5, 5, 2 times matrix D, which is all of this. So we're going to multiply it times 3, 3, 4, 4, negative 2, negative 2. Now the first thing that we have to check is whether this is even a valid operation.

Now the matrix multiplication is a human-defined operation that just happens-- in fact all operations are-- that happen to have neat properties.

Now the way that us humans have defined matrix multiplication, it only works when we're multiplying our two matrices. So this right over here has two rows and three columns.

So it's a 2 by 3 matrix. And this has three rows and two columns, it's 3 by 2. This only works-- we could only multiply this matrix times this matrix, if the number of columns on this matrix is equal to the number of rows on this matrix.

And in this situation it is, so I can actually multiply them. If these two numbers weren't equal, if the number of columns here were not equal to the number of rows here, then this would not be a valid operation, at least the way that we have defined matrix multiplication. The other thing you always have to remember is that E times D is not always the same thing as D times E. Order matters when you're multiplying matrices.

It doesn't matter if you're multiplying regular numbers, but it matters for matrices. But let's actually work this out. So what we're going to get is actually going to be a 2 by 2 matrix. But I'm going to create some space here because we're going to have to do some computation. So this is going to be equal to-- I'm going to make a huge 2 by 2 matrix here.

So the way we get the top left entry, the top left entry is essentially going to be this row times this product. If you view them each as vectors, and you have some familiarity with the dot product, we're essentially going to take the dot product of that and that.

And if you have no idea what that is, I'm about to show you. This entry is going to be 0 times 3, plus 3 times 3, plus 5 times 4. So that is the top left entry. And I already see that I'm going to run out of space here, so let me move this over to the right some space so I have some breathing room. Now we can do the top right entry. This was the top left, now we're going to do the top right.

So the top right entry is going to be this row times this column. Notice the entry is getting the row from the first matrix and the column from the second one. That's kind of determining its position. So, once again, is going to be 0 times 4, plus 3 times negative 2, plus 5 times negative 2.