Answer each question. What is the product [3x3 matrix] [3x3 matri...

Question

Answer each question. What is the product [3x3 matrix] [3x3 matrix]?

Accepted Answer

Hello, today we're going to be performing the multiplication of the given matrices. So what we are given is the three by three matri C of negative 32 negative 49, 11 8, 10 0 and eight. And we're gonna be multiplying it with the second matrix which is 100010 and 001. So the question is what is going to be the shape of our resulting matrix? Well, in order to check, we can set up the dimension check, we are multiplying a three by three matrix with another three by three matrix, we need to check to see if these two middle numbers are matching and since three and three are matching, that means that the multiplication is going to be possible. So the dimensions of the resulting matrix are going to be the outer numbers of the parameters which in this case is going to be a three by three. So we know that the remaining matrix after the multiplication is going to be of the form of a three by three. So how do we carry out the matrix multiplication? Well, in order to get the first element of our first new matrix, we have to take the first element in the first row of the first matrix and multiply it to the first column of the second matrix that is going to give us negative three times one, which is negative three plus negative three times zero, which is zero plus negative three times zero, which is zero. Then we're gonna repeat this process for the next element in the first row. We're gonna be taking positive nine and multiplying it to the second row for our next element. I'm sorry, the second column. So we're going to get nine times zero, which is zero plus nine times one, which is nine plus nine times zero, which is zero. Then moving on, we're going to take the third element of the first row which is 10 and multiply it into the third column. So that is going to give us 10 times zero, which is zero plus 10 times zero, which is zero plus 10 times one, which is 10. Now we're gonna move on to our second row of the new matrix. In order to get the second row, we're going to take the first element of the second row of the first matrix and multiply it to the first column that is going to give us two times one, which is two plus two times zero, which is zero plus two times zero, which is zero. Then moving on, we take the next element which is 11. And we're gonna multiply it to the second column of the second matrix that is going to give us 11 times zero, which is zero plus 11 times one, which is 11 plus 11 times zero, which is zero. Next, we're going to take the next element which is zero. And we're gonna multiply it to e the third column in the third matrix, but zero times any number is just going to give us zero. So that new element is going to be zero. Finally, what we're going to do for our third row is we're going to take the first element of the third row of the first matrix and multiply it to the first column in the second row that is going to give us negative four times one, which is negative four plus negative four times zero, which is zero plus negative four times zero, which is zero. Then we're gonna take the next element which is positive eight and multiply it to the second column that is going to give us eight times zero, which is zero plus eight times one, which is eight plus eight times zero, which is zero. And finally, for our last element, we're going to be taking the next element which is eight and multiplying it to the third column in the second matrix that is going to give us eight times zero, which is zero plus eight times zero, which is zero plus eight times one, which is eight. So now that we have all the elements written now for our new matrix, we just need to go ahead and simplify each of them. So for the first element in the first column, we have negative three plus zero plus zero, which is negative three for the second row. Second element in the first row, we have zero plus nine plus zero, which is going to give us nine. And for the third element in the first row, we have zero plus zero plus 10, which is just going to be 10. Now, for the first element in the second row, we have two plus zero plus zero, which is just two. For the next element we have zero plus 11 plus zero, which is 11, then we have zero. And for the final row, we have negative four plus zero plus zero, which is negative 40 plus eight plus zero, which is eight and zero plus zero plus eight, which is going to be eight. And this is going to be the result of the matrix multiplication. And with that being said, the answer to this problem is going to be D. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Answer each question. What is the product [3x3 matrix] [3x3 matrix]?

Key Concepts

Matrix Multiplication

Dimensions of Matrices

Dot Product

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