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[M] Construct a random $4 \times 4$ matrix $A$ with integer entries between $-9$ and $9 .$ How is det $A^{-1}$ related to det $A ?$ Experiment with random $n \times n$ integer matrices for $n=4$ 5, and $6,$ and make a conjecture. Note: In the unlikely event that you encounter a matrix with a zero determinant, reduce it to echelon form and discuss what you find.

$\operatorname { det } ( \mathrm { A } ) = 1 / \operatorname { det } \left( A ^ { - 1 } \right)$ Proved

Algebra

Chapter 3

Determinants

Section 1

Introduction to Determinants

Introduction to Matrices

McMaster University

Harvey Mudd College

Baylor University

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

01:39

[M] Construct a random $4 …

01:30

$[\mathbf{M}]$ If det $A$ …

01:24

For a randomly generated $…

01:40

Conjecture Consider square…

00:47

Let $A$ and $B$ be $4 \tim…

02:38

00:37

Find the determinant of th…

02:58

Consider square matrices i…

01:29

02:12

Okay. So in this example, we want to see how these four transformations will affect the determines off A when we use random former for matrices. So I created a love another program here, where we will generate random matrices depending on this shape. Yeah, so if you want to use it on five by five and 656 matrix Just changed history. Five comma five and six. Common six. Then what this will do is that this will find the ratio between eight transfers on A. So it's going to wipe transfers and eight to the ratio between negative, eh? And eh, The ratio between two. A two and ratio of 10 8 to a and then everything worry about what's called us. But that's what it does. So what you find out is that the determined a transpose should be equal to two determinant off, eh? Determined off negative. A is equal to negative one times by and determines all, eh? So here I'm just gonna make a random a general n by N matrix. And then this one will be true to the end determinants. Oh, hey, this is 10 to the end tournaments away and in this case because this is for for so many years, for this will just be one to the end. So it's gonna be equal to determine off A is going to be too to the form, so you'll get 16 and this is tentative. Fall and you'll you'll see this when you perform this program.

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Missouri State University

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

[M] Construct a random $4 \times 4$ matrix $A$ with integer entries between …

$[\mathbf{M}]$ If det $A$ is close to zero, is the matrix $A$ nearly singula…

For a randomly generated $5 \times 5$ matrix, verify that $\operatorname{det…

Conjecture Consider square matrices in which the entries are consecutive int…

Let $A$ and $B$ be $4 \times 4$ matrices such that $\operatorname{det}(A)=5$…

Find the determinant of the matrix.$$\left[\begin{array}{rr}9 & \sqr…

Consider square matrices in which the entries are consecutive integers. An e…

$[\mathbf{M}]$ Let $H=\operatorname{Span}\left\{\mathbf{v}_{1}, \mathbf{v}_{…

00:50

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

00:30

Compute the determinants of the elementary matrices given in Exercises $25-3…

02:56

Describe the possible echelon forms of the matrix. Use the notation of Examp…

02:48

Repeat Exercise 5 with $C=\left[\begin{array}{cc}{.1} & {.6} \\ {.5} &am…

01:34

Each equation in Exercises $1-4$ illustrates a property of determinants. Sta…

02:29

Suppose $A$ is an $m \times n$ matrix with the property that for all $\mathb…

01:27

Determine which sets in Exercises $15-20$ are bases for $\mathbb{R}^{2}$ or …

02:28

For each subspace in Exercises 1–8, (a) find a basis, and (b) state the dime…

02:25

$[\mathbf{M}]$ With $A$ and $B$ as in Exercise $41,$ select a column $\mathb…

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