In Problems 25-30, find the product. [ 1 0 1 2 4 1 3 6 1

step 1 Hello. So here we're finding the product of these two matrices. So notice that our first matrix,

In Problems 25-30, find the product. [ 1 0 1 ...

Key Concepts

Dot Product

The dot product is the process used in matrix multiplication where each element in the resulting matrix is obtained by multiplying corresponding entries from a row of the first matrix and a column of the second matrix, then summing these products. It is fundamental in computing the elements of the product matrix.

Resulting Matrix Dimensions

The dimensions of the resulting matrix from a multiplication are given by the number of rows in the first matrix and the number of columns in the second matrix, which are derived from the requirement of compatibility in matrix multiplication.

Matrix Multiplication

Matrix multiplication is a binary operation that takes a pair of matrices and produces another matrix. This operation involves multiplying corresponding elements and summing them according to a specific rule, which is essential in many areas of applied mathematics and engineering.

Compatibility of Dimensions

For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. This requirement ensures that the multiplication is defined and leads to a resulting matrix whose dimensions are determined by the number of rows from the first matrix and the number of columns from the second matrix.

Transcript

0:00 Hello.

00:01 So here we're finding the product of these two matrices.

00:04 So notice that our first matrix, we have three rows and three columns.

00:09 So our first matrix is a three by three matrix.

00:14 And our next matrix is a, well, we have three rows and two columns.

00:18 Therefore, we have a three by two matrix.

00:23 Now notice that the inner numbers match up.

00:24 The number of columns of the first matrix matches the number of rows of the second matrix.

00:29 So therefore, we can actually.

00:30 Do multiplication and what we're going to get is the outer numbers we're going to get a three by two matrix out of this so we know that we have a three by two matrix so for our first element or in the first row first column we do the dot product of the first row with the first column that's one times one plus zero times six plus one times eight so that's going to give us um um again one plus plus 0 plus 8, which gives us 9.

01:08 So the first element here is going to be 9.

01:11 Okay, so now let's go to the first row, second column.

01:15 So then that's the dot product, first row with the second column.

01:18 So that's going to be 1 times 3 plus 0 times 2 plus 1 times negative 1.

01:22 So 3 plus 0 minus 1, which is 2.

01:27 So the element there is 2.

01:29 Okay, now let's go to the second row first column.

01:32 So second row, first column.

01:34 So again, the dodd product here is two times one plus four times six plus one times eight.

01:40 So two plus, do you do, where we're at? we're the second row first column.

01:53 So again, two plus 24 plus eight.

01:57 So that's going to give us 34...