1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Alex_Xolod [135]
3 years ago
8

In matrix multiplication, such as AB=C, the columns of B form the rows of C. why is this?

Mathematics
1 answer:
Gnesinka [82]3 years ago
5 0
Let's work with 2-by-2 matrices so we're on the same page. The ideas will work for any appropriate matrices.

From the rule of matrix multiplication, we see:
\left[\begin{array}{cc}a_{11} & a_{12} \\a_{21} & a_{22} \end{array}\right] \left[\begin{array}{cc}b_{11} & b_{12} \\b_{21} & b_{22} \end{array}\right] = \left[\begin{array}{cc} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22} b_{22} \end{array}\right]
As you noted, we see the columns of B contributing to the rows of C. The question is, why would we ever have defined matrix multiplication this way?

Here's a nontraditional way of feeling this connection. We can define matrix multiplication as "adding multiplication tables." A multiplication table is made by starting with a column and a row. For example,
\begin{array}{ccc} {} & 1 & 2 \\ 1 & {} & {} \\ 2 & {} & {} \end{array}
We then fill this table in by multiplying the row and column entries:
\begin{array}{ccc} {} & [1] & [2] \\ 1| &1 & 2 \\ 2| & 2 &4 \end{array}
It's then reasonable to say that given two matrices A and B, we can construct multiplication tables by taking the columns of A and pairing them with the rows of B:
\left[\begin{array}{cc}a_{11} & a_{12} \\a_{21} & a_{22} \end{array}\right] \left[\begin{array}{cc}b_{11} & b_{12} \\b_{21} & b_{22} \end{array}\right]

= \begin{array}{cc} {} & \left[\begin{array}{cc} b_{11} & b_{12}\end{array} \right]\\ \left[\begin{array}{c} a_{11} \\ a_{21} \end{array} \right] \end{array} +\begin{array}{cc} {} & \left[\begin{array}{cc} b_{21} & b_{22}\end{array} \right]\\ \left[\begin{array}{c} a_{12} \\ a_{22} \end{array} \right] \end{array}

= \left[\begin{array}{cc} a_{11} b_{11} & a_{11} b_{12} \\ a_{21} b_{11} & a_{21} b_{12} \end{array} \right] + \left[\begin{array}{cc} a_{12} b_{21} & a_{12} b_{22} \\ a_{22} b_{21} & a_{22} b_{22} \end{array} \right]

Adding these matrices together, we get the exact same expression as the traditional definition. 




You might be interested in
this is pretty easy to most people its just that ive never studied this (open attachment) im in year 7
yKpoI14uk [10]

Answer:

a. 8/20 - 2/5

b. 2/8

Step-by-step explanation:

6 0
2 years ago
Read 2 more answers
Sound travels through sea water at a speed of about 1500 meters per second. At this rate, how far will sound travel in 2 minutes
Mariulka [41]

Answer:

180,000 m

Step-by-step explanation:

There are 60 seconds in a minute, so:

2 min × 60 s/min = 120 s

Distance = rate × time

d = 1500 m/s × 120 s

d = 180,000 m

8 0
3 years ago
Hazel has band practice every other day and volunteers every seven days. if she had a band practice and volunteered on oct 23, w
LuckyWell [14K]
Cjdfb hjkbftbbui yuftg
3 0
3 years ago
What is Mary's running speed in<br> miles per hour?
AysviL [449]

Answer:

4 to 6 miles an hour would be the answer

4 0
3 years ago
Read 2 more answers
Draw a model that shows 2 items being divided equally for 4 people. How much will each person have?
zhenek [66]

Answer:

hello friend!

Step-by-step explanation:

A half or 1/2

if i  have 2 apples and 4 friends ofc i donr have enough for each friend to get a whole or one apple so to make it fair, i would have to split each apple so each person could get a half

3 0
2 years ago
Read 2 more answers
Other questions:
  • Find the discount if the original price is $8.05 and the percent markdown is 10 % round to the nearest cent
    13·2 answers
  • Factor the polynomial completely.<br> (y - 8)^2 - 16
    14·1 answer
  • 1 of every 20 cookies falls apart. if you make 40 cookies how many will fall apart? 5 1 3 2
    14·1 answer
  • Please can someone help me
    14·1 answer
  • Consider the sequence: 3, 8, 13, 18, 23, ...The recursive formula for this sequence is:an = an-1 + 5In a COMPLETE sentence, expl
    6·1 answer
  • 2 fair dice are rolled. what is the probability that the sum is even given that the first die is rolled is a 2?
    8·2 answers
  • 7/2x + 1/2x = 9 + 9/2x<br> the value of x is
    8·1 answer
  • Help me solve this question
    6·1 answer
  • Need help with geometry (number 19)
    10·1 answer
  • -6 + 6 how do you solve it
    7·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!