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Show that $A I_{n}=A$ when $A$ is an $m \times n$ matrix. [Hint: Use the (column) definition of $A I_{n} . ]$

$A I_{n}=A$

Algebra

Chapter 2

Matrix Algebra

Section 1

Matrix Operations

Introduction to Matrices

Missouri State University

Campbell University

Oregon State University

Lectures

01:32

In mathematics, the absolu…

01:11

08:16

Let $A$ be an $m \times n$…

04:14

Let $\mathbf{A}$ be an $n …

05:50

Identity Matrix Let $A=\le…

02:46

Show that $I_{m} A=A$ when…

04:04

Prove that if $A$ is an $m…

02:13

01:08

Suppose that $D$ is an $n …

02:31

Let $A$ be an $n \times n$…

03:50

Prove that if $A$ is an $n…

03:00

okay for question 32. Here we want to show that Yeah. Matrix a times The n identity matrix is equal to a when a is an m upset. Getting ahead of myself is an M by n matrix on so matrix a times the end identity matrix by the definition of matrix multiplication the same thing as a times And now I'm going to write Oh, actually, yeah. Sorry is going Teoh, right? It the wrong way. We have e one as the basis the first basis of our in dimensional space. So that would be the 10000 etcetera Vector to up Thio n by the definition of matrix multiplication that is the same thing as A and e one. A time to eat too, all the way up to a 10 e n. Now, if we consider one of those columns E one Well, that's the same thing is a times 1000 dot, dot dot Which when we do our matrix are multiplication of the Matrix times. This vector we would get that is a 11 a 21 a 31 up to eight times and one We're multiplying the first row element or sorry, The first column element of each row by one and every other column element by zero and similarly e N is just going to end up being a n one. A n two dot dot, dot dot hopes. You should write this down to being not a n n So our matrix a tends the identity matrix just going to give us oh, and also actually specify that is the same thing as our A one vector. And that down there is the same thing as R A N Factor and every other one in between will be similar. We have a eight times e one going to be our A one vector. Our first column vector of a eight times that you two is going to be our second column Vector of a and eight times e N is going to be our 10th column vector of a, which by definition, that's just Hey, don t o g m matrix bracket. By definition, that's just a

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