Multiplying a matrix by another matrix

Multiplying a matrix by another matrix

In this lesson, we will learn how to multiply a matrix with another matrix. But we will learn about n-tuples first. An n-tuple is an ordered list of n numbers. Multiplying an n-tuple by another n-tuple is called the dot product. The dot product is the summation of all product of each corresponding entries. To multiply a matrix with another matrix, we have to think of each row and column as a n-tuple. Each entry will be the dot product of the corresponding row of the first matrix and corresponding column of the second matrix. For example, if your entry is at the 3rd row and 4th column, then you have to take the dot product of the 3rd row of the first matrix and 4th column of the second matrix. Note that not all matrices can be multiplied.

Lessons

Notes:
In this section we will learn how to multiply two matrices like A \cdot B together.

A nn-tuple is an ordered list of nn numbers. For example,
(1,2,3,4) is an ordered quadruple with 4 numbers, and (1,2,3) is an ordered triple with 3 numbers. We usually specify each ordered nn-tuple as a variable with an arrow on top. For example,

x=(1,2,3,4)\vec{x}=(1,2,3,4)

If we have 2 ordered nn-tuples, then we can find the dot product. The dot product is summation of all the product of each corresponding entries. For example, let

a=(1,2,3)\vec{a}=(1,2,3) and b=(2,2,2)\vec{b}=(2,2,2).

If we do the dot product of a\vec{a} and b\vec{b} and ,then we will get the following:

ab=(1,2,3)(2,2,2)\vec{a}\cdot\vec{b}=(1,2,3)\cdot(2,2,2)
=12+22+32=1\cdot2+2\cdot2+3\cdot2
=2+4+6=2+4+6
=12=12

When we want to multiply a matrix by a matrix, we want to think of each row and column as a nn-tuple. To be exact, we want to focus on the rows of the first matrix and focus on columns of the second matrix. For example,



For example, r1\vec{r_1}is the first row of the matrix with an ordered triple (1,2,3). Now to multiply these two matrices, we need to use the dot product of r1\vec{r_1} to each column, dot product of r2\vec{r_2} to each column, and r3\vec{r_3} to each column. In other words



    • a)
      Dot product
    • b)
      Multiplying a matrix by another matrix
  • 2.
    Dot product
    Find the dot product of the following ordered nn-tuples:
  • 3.
    Multiplying matrices
    Multiply the following matrices:
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Multiplying a matrix by another matrix

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