Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Use the algorithm from this section to find the inverses of $$\left[\begin{array}{lll}{1} & {0} & {0} \\ {1} & {1} & {0} \\ {1} & {1} & {1}\end{array}\right] \text { and }\left[\begin{array}{llll}{1} & {0} & {0} & {0} \\ {1} & {1} & {0} & {0} \\ {1} & {1} & {1} & {0} \\ {1} & {1} & {1} & {1}\end{array}\right]$$

When we multiply any other row from $A$ with column from $B$ result is zero.

Algebra

Chapter 2

Matrix Algebra

Section 2

The Inverse of a Matrix

Introduction to Matrices

Campbell University

McMaster University

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

09:52

Find the inverse of the gi…

01:04

Find the inverse, if it ex…

05:27

02:23

Finding the Inverse of a M…

01:37

10:27

01:46

Use a calculator to find $…

01:39

Find the inverse of each m…

03:06

Determine whether $B$ is t…

02:31

In this example, we have obey tricks A, which is given to be lower triangular and the entries in the lower triangular portion are all exactly one. This matrix will be in vertebral and so let's calculate its inverse. So to do that are algorithm is to make a matrix formed by a and augmented by the three by three identity matrix. So this is going to be equal to first let me place a in blue. It's 100 110 and then 111 Then we've augmented with the three by three identity matrix, which is 100 010 and 001 So this is our matrix, and our job is to roll reduce until this portion before being augmented becomes the identity matrix. So now to that end, we have a pivot here, and our operations will be to eliminate these two entries. So first we copy Row one, it's 100 100 Then take negative one times Row one and add it to the second row will attain 010 negative one, one and zero and for the next operation will copy multiply the first row by negative one and added to road to we'll obtain this time 01 one. You obtain a negative one zero and one altogether. Now, for this new matrix, we have a pivot. Here we have a zero above, which is good, but we need to eliminate this entry, making it a zero. So will say that this is roll equivalent to first copy rose one into So we have 100100 and row one row two is 010 negative. 110 And now our operation will be once again Multiply wrote to buy a negative one and added to row three will obtain 001 Then we have a zero here than a negative one and a positive one. So now let's bring back the fact that we've augmented. Since this is I three, we've now shown that a inverse exists and it's equal to 100 negative. 110 and zero negative one and one. Then what you can do to check that we really found the inverse compute a times a inverse and show that you get the three by three identity matrix. Let's redo this example, where instead of a three by three upper triangular matrix consisting of ones, let's make it be a four by four case. So let's see what the inverse will be in this situation. Well, this is a tricky matrix, not because it's going to be overly difficult to take a augment with the four by four Identity Matrix and ro reduce. It's just a very long process. And when we make many, many different with medical operations, the probability of one tiny mistake is quite high. So let's go back to the Matrix we looked at in the three by three case. Recall that we had this interesting form. Row one is 100 Let's go back to our case and hope that that still works out. Let's say for a inverse our row one will be one, then zeros. It's of size four by four. So it'll be about this large. Okay, let's pop back to a inverse and note that in the second row what we've done is we have negative one, then one and then zeros thereafter. So let's make it negative one, then one in our case here and zeros thereafter, hoping that this works. So let's go back to the three by three case and notice. Now. In the last row, we had zero than a negative one and a positive one. Let's maybe use a different highlighting color. So let's try that out for our case. Let's go. Zero negative one and one in the four by four. So zero negative one and one. And I hope maybe a zero could go here. Now we have kind of a staircase pattern. It goes negative 11 Then go down a row. Negative 11 And my hope is there's gonna be a negative one and one here, Let's try it out. Place a negative one and one and zeros elsewhere. So this is our form for the Matrix. And I've never did roll reduction to verify this. We just looked at the pattern that worked out in the three by three case. Now for this matrix A provided here and the a inverse we've just calculated you can check that when we multiply. We, in fact get the four by four identity matrix that tells us our guests is correct

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 …

Find the inverse of the given matrix.$$\left[\begin{array}{llll}{1} &…

Find the inverse, if it exists, for each matrix.$$\left[\begin{array…

Find the inverse of the given matrix.$$\left[\begin{array}{llllll}{1} &a…

Finding the Inverse of a Matrix Find the inverse of the matrix if it exists.…

Find the inverse, if it exists, for each matrix.$$\left[\begin{array}{rr…

Find the inverse of the given matrix.$$\left[\begin{array}{rrrr}{-1} &am…

Use a calculator to find $A^{-1}=B,$ then confirm the inverse by showing $A …

Find the inverse of each matrix.$$\left[\begin{array}{lll}1 & 0 &…

Determine whether $B$ is the multiplicative inverse of $A$ using $A A^{-1}=I…

05:25

Suppose $T$ and $U$ are linear transformations from $\mathbb{R}^{n}$ to $\ma…

03:08

[M] With $D$ as in Exercise 41 , determine the forces that produce a deflect…

04:20

Find an LU factorization of the matrices in Exercises $7-16$ (with $L$ unit …

02:59

$[\mathbf{M}]$ Let $\mathbf{b}=\left[\begin{array}{l}{7} \\ {5} \\ {9} \\ {7…

04:14

If $A$ is an $n \times n$ matrix and the equation $A \mathbf{x}=\mathbf{b}$ …

01:49

Exercises 21 and 22 concern the way in which color is specified for display …

03:09

Exercises $17-20$ refer to the matrices $A$ and $B$ below. Make appropriate …

04:12

Suppose $A=B C,$ where $B$ is invertible. Show that any sequence of row oper…

02:41

In a certain region, about 6% of a city’s population moves to the surroundin…

05:54

Let $\mathbf{v}_{1}=\left[\begin{array}{r}{2} \\ {3} \\ {-5}\end{array}\righ…

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.