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Suppose that all the entries in $A$ are integers and det $A=1 .$ Explain why all the entries in $A^{-1}$ are integers.

since each entry of adjA is an integer, entries in the inverse of $A$ are also integers.

Algebra

Chapter 3

Determinants

Section 3

Cramer’s Rule, Volume, and Linear Transformations

Introduction to Matrices

Campbell University

Baylor University

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

01:37

Suppose that all the entri…

02:08

03:55

02:21

01:45

CHALLENGE If $a_{1}$ and $…

00:27

Explain why the whole numb…

01:49

Show that if $a | b$ and $…

00:30

In your own words, explain…

08:26

Show that if $a$ and $b$ a…

01:55

For what values of $a$ is …

in this video, we're gonna be going over how to solve problem number 18 from section 2.3. Wishes on Cramer's rule in determines. So the question because a prompt that, um and says that the Matrix has and has all entries to be integers so like whole numbers. Um, and the determinant of a is one. And it s says to explain why all the entries in the inverse of a have to be introduced as well. So Thio do that. It's pretty simple. We just consider this formula that was given to us and they were made, Which is that that the inverse inverse of matrix is one over the determinant of the Matrix times, the advocate of the Matrix. So you scale every entry in the as yet of the Matrix by the determinant. So, um, if we have like, let's say, three by three matrix with nine entries, The advocate of the Matrix is no the cool factor of every entry transposed in a matrix. So if you want to find a co factor for that, uh, for the 1st 0 for example, you basically block or like cross out the row and column that contains that entry and take the German of the remaining two by two. And you repeat that for every single element and then transpose that and put it in the matrix in the same spot and transpose it. And they're actually alternating signs of plus minus minus, plus minus plus minus. Plus. When you take the determinants, you should have a plus or minus sign based on what entry it is corresponding to it. So that's how you find an adjective, a matrix. So basically, this just involves taking the determinant of a two by two, which is basically, uh, the diagonals multiplied by each other and take their difference. So when you multiply indigenous together or subtract, add them, they'll still be integers. You'll end up with integers. Therefore, the inverse is going to be scaled by one over one because the determinant of a is given to be one times the advocate of a. So the inverse is just the as you go today and since we know that the advocate of a contains only integers, because taking a determinant will give us a whole number. If we use all integers, then we also know that every entry and the advocate of a or the inverse of

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