Multiplying a matrix by another matrix

Multiplying a matrix by another matrix

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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