In Problems 25-30, find the product. [ 1 -1 -3 2 0 5 ][ 2 8

step 1 Hello, so here we are multiplying these two matrices. So our first matrix is a 3 by 2 matrix. We

In Problems 25-30, find the product. [ 1 -1 -3...

Key Concepts

Dot Product

The dot product is a fundamental operation used in matrix multiplication. It involves multiplying corresponding entries of two sequences of numbers (typically a row from the first matrix and a column from the second matrix) and summing the products to obtain a single number, which becomes an entry of the resulting matrix.

Matrix Dimensions and Multiplicability

The concept of matrix dimensions is crucial because it determines whether two matrices can be multiplied together. For the multiplication A * B to be defined, the number of columns in matrix A must match the number of rows in matrix B, resulting in a product matrix whose dimensions are the number of rows of A by the number of columns of B.

Matrix Multiplication

Matrix multiplication is a method of combining two matrices to produce a new matrix, where each element of the resulting matrix is obtained by taking the dot product of a row from the first matrix and a column from the second matrix. This operation is defined only when the number of columns in the first matrix is equal to the number of rows in the second matrix.

Transcript

00:01 Hello, so here we are multiplying these two matrices.

00:04 So our first matrix is a 3 by 2 matrix.

00:07 We have 3 rows and 2 columns.

00:09 Let's say 3 by 2 by 2 matrix.

00:11 We're multiplying by a 2 by 3 matrix where we have 2 rows and 3 columns.

00:16 So this is going to be a 2 by 3 matrix.

00:19 And we see here that the inner numbers match up.

00:22 In other words, the columns of the first matrix matches the rows of the second matrix.

00:28 So the multiplication here is defined.

00:29 What we're going to get is a three by three matrix.

00:33 So we know that we have a three by three matrix that what we're going to get when we do the application.

00:37 And how we just do this is we do the dot product with the ith row of the first matrix with the jeth column for our, you know, our element that is in the, you know, the i .j position.

00:52 So what does that mean? it basically, okay, the first element here is going to be what we do is our first row, first column.

01:00 So we do the dot product with the first row with the first column.

01:04 So that's going to be 1 times 2 and then plus negative 1 times 3, right? the first row with the first column.

01:15 So 1 plus 2 plus 1 times 2 plus negative 1.

01:20 That gives us negative 1.

01:23 So our first element is negative 1.

01:25 And then we go to the first row second column.

01:27 So that's going to be first row.

01:29 First row second column so that's one times eight plus negative one times six so it's eight um plus negative six or eight minus six which is two and then continuing we have our first row third column so one times negative one plus negative one times zero so that's minus one plus zero one minus one and it's continuing now we go to the second row first column so second row first column...

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