💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Watch this step-by-step video, matched to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Prove Theorem 3$(\mathrm{d}) .\left[\text { Hint: Consider the } j \text { th row of }(A B)^{T} .\right]$

$(A B)^{T}=B^{T} A^{T}$

Algebra

Chapter 2

Matrix Algebra

Section 1

Matrix Operations

Introduction to Matrices

Campbell University

McMaster University

Lectures

01:32

In mathematics, the absolu…

01:11

04:26

Prove Theorem 2$(\mathrm{b…

06:06

Prove Theorem 2$(\mathrm{d…

08:27

Prove Theorem 3 as follows…

04:06

(a) Prove Theorem 4, part …

02:33

Prove Formula 3 of Theorem…

04:42

Prove Theorem 3

15:11

03:27

Proof Prove Property 3 of …

04:27

Provea) part (ii) of T…

01:22

Prove that part (iii) of T…

in this example, we have two. Major sees The Matrix A is going to use the notation a i J for its entries. Then B i j will be the entries of the Matrix B. We're going to be looking at the product a Times B. So we also have the definition of what the I. J entry of that product would look like. It would be this some in particular. So now let's state what our goal is for this particular video. We want to prove that if we take a times B and transpose the results, then this is equal to be transposed times a transposed. So it's the product of the matrices in reverse order, both being transposed. I think this is a pretty surprising result, personally, so let's give this proof a try. Let's start by considering the arbitrary I J element of eight times be transposed the left hand side. So will say that the I J entry off eight times be transposed is of this form. I'll use this notation to refer to that entry. First, I'll put a times B transposed and a subscript I J. So we do know that when we take a times B transposed and we're looking for the I J entry of it. It's the same as looking at the product a Times B. J. I entry since the transposed has effective inter changing rows and columns, so equivalently the interchanges expressed here by switching those two entries. So it's right out next using this formula what this would be equal to when we use that formula. The major thing toe watch out for is just the switch in the indices so we can begin by writing that this is not a I j but a J I or J one times not be one j, but rather be one I then like light. Likewise, we have a J two times be too I plus all the way down the list until we get to a J N Times B and I. So switching the indices had the effective switching this I to make it a j and this J to make it I and then similarly for the rest of the products. So now let's look at the right hand side. We're going to focus on be transposed times a transposed next, and the strategy is going to be to consider the I J entry. That's a J there of be transposed times a transposed. If we can determine what the's entries are and find that there same as thes entries, then this proves equality. So let's use the notation that be transposed. A transpose i j entry will be equal to the following. What we're going to do is go back to the equation for matrix multiplication. But now just use the fact that becomes first also be has been transposed, So we won't use be i. J. I won Here will stand use be one j since transposed switches the entries and we're using B because the matrix B comes first here. So this will then be equal to be one j times A or excuse me will be be one i times a J one. Then go to the next term. It will be be too. I times a j two. Then the very last entry of this product is going to be be and I times a J. And that's a lot of work with double indices. But that's the going to be the I J entry of be transposed times a transpose Now let's carefully look at these entries and compare him to these notice that the only difference between these entries is the order of the products. For example, here I have be one J and here Apia have be one J. It comes as either the second factor or the first factor, but all these terms are equivalent. And so we have just shown that the I J entry of a be transposed is the same as the I J entry of be transposed times. Eight transposed. So it's give our conclusions. We can say, since the I J entry of eight Times Be transposed is equal to the I J entry of be transposed times. A transposed it follows that eight times be transposed is equal to be transposed times a transposed as required.

View More Answers From This Book

Find Another Textbook

Numerade Educator

In mathematics, the absolute value or modulus |x| of a real number x is its …

Prove Theorem 2$(\mathrm{b})$ and 2$(\mathrm{c}) .$ Use the row-column rule.…

Prove Theorem 2$(\mathrm{d}) .$ [Hint: The $(i, j)$ -entry in $(r A) B$ is $…

Prove Theorem 3 as follows: Given an $m \times n$ matrix $A,$ an element in …

(a) Prove Theorem 4, part 3.(b) Prove Theorem 4, part 5.

Prove Formula 3 of Theorem 3.

Proof Prove Property 3 of Theorem $1.1 .$ (You may use Property 3 of Theorem…

Provea) part (ii) of Theorem 4 .b) part (iii) of Theorem 4.

Prove that part (iii) of Theorem 1 is true.

02:01

Determine by inspection whether the vectors are linearly independent. Justif…

02:45

$[\mathbf{M}]$ Find a column of the matrix in Exercise 40 that can be delete…

01:19

In Exercises 29–32, find the elementary row operation that transforms the fi…

02:18

In the rest of this exercise set and in those to follow, you should assume t…

09:21

a. Compute the transfer matrix of the network in the figure.b. Let $A=\l…

01:09

How many rows does $B$ have if $B C$ is a $3 \times 4$ matrix?

02:12

In Exercises 11 and $12,$ give integers $p$ and $q$ such that Nul $A$ is a s…

03:26

Column vectors are written as rows, such as $\mathbf{x}=\left(x_{1}, x_{2}\r…

Compute the determinants in Exercises $9-14$ by cofactor expansions. At each…

00:51

Compute the determinants in Exercises $1-8$ using a cofactor expansion acros…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.