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Lubov Fominskaja [6]
3 years ago
13

What would you have to know about the pivot columns in an augmented matrix in order to know that the linear system is consistent

and has a unique solution?
Computers and Technology
1 answer:
Scrat [10]3 years ago
8 0

Answer:

The Rouché-Capelli Theorem. This theorem establishes a connection between how a linear system behaves and the ranks of its coefficient matrix (A) and its counterpart the augmented matrix.

rank(A)=rank\left ( \left [ A|B \right ] \right )\:and\:n=rank(A)

Then satisfying this theorem the system is consistent and has one single solution.

Explanation:

1) To answer that, you should have to know The Rouché-Capelli Theorem. This theorem establishes a connection between how a linear system behaves and the ranks of its coefficient matrix (A) and its counterpart the augmented matrix.

rank(A)=rank\left ( \left [ A|B \right ] \right )\:and\:n=rank(A)

rank(A)

Then the system is consistent and has a unique solution.

<em>E.g.</em>

\left\{\begin{matrix}x-3y-2z=6 \\ 2x-4y-3z=8 \\ -3x+6y+8z=-5  \end{matrix}\right.

2) Writing it as Linear system

A=\begin{pmatrix}1 & -3 &-2 \\  2& -4 &-3 \\ -3 &6  &8 \end{pmatrix} B=\begin{pmatrix}6\\ 8\\ 5\end{pmatrix}

rank(A) =\left(\begin{matrix}7 & 0 & 0 \\0 & 7 & 0 \\0 & 0 & 7\end{matrix}\right)=3

3) The Rank (A) is 3 found through Gauss elimination

(A|B)=\begin{pmatrix}1 & -3 &-2  &6 \\  2& -4 &-3  &8 \\  -3&6  &8  &-5 \end{pmatrix}

rank(A|B)=\left(\begin{matrix}1 & -3 & -2 \\0 & 2 & 1 \\0 & 0 & \frac{7}{2}\end{matrix}\right)=3

4) The rank of (A|B) is also equal to 3, found through Gauss elimination:

So this linear system is consistent and has a unique solution.

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nikdorinn [45]

Answer:

#include <iostream>

#include<time.h>

using namespace std;

void sort(int a[20],int n)   //selection sort

{

   int i,j,temp;

   for(i=0;i<n;i++)        

   {

       for(j=i+1;j<n;j++)  //finds minimum element

       {

           if(a[i]>a[j])     //swaps minimum element to its right place

           {

               temp=a[i];

               a[i]=a[j];

               a[j]=temp;

           }

       }

   }

}

int main()

{

   srand(time(NULL));

   int arr[20],n=20,i;

   for(i=0;i<n;i++)

       arr[i]=rand() % 100 + 1;    //assigns random number to each array element

   cout<<"Before sorting\n";

   for(i=0;i<n;i++)

       cout<<arr[i]<<" ";   //prints array before calling sort func

   sort(arr,20);     //sort the passed array

   cout<<"\nAfter sorting\n";

   for(i=0;i<n;i++)

       cout<<arr[i]<<" ";  //prints array after sorting

   return 0;

}

OUTPUT :

Please find the attachment below.

Explanation:

The program above first assigns random number from 1 to 100 to each element of array using rand(). srand(time(NULL)) is used to generate new number every time otherwise same random number will be produced at a time. Then sorting method is called in which selection sort algorithm is used.

Selection sort algorithm finds the minimum element in each pass and place it into its right position. For example - 34,54,12,5 is passed

After Pass 1 - 5  54  12  34

After Pass 2 - 5  12  54  34

After Pass 3 -  5  12  34  54

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3 0
3 years ago
Write c++ program to find maximum number for three variables using statement ?​
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Answer:

#include<iostream>

using namespace std;

int main(){

int n1, n2, n3;

cout<<"Enter any three numbers: ";

cin>>n1>>n2>>n3;

if(n1>=n2 && n1>=n3){

cout<<n1<<" is the maximum";}

else if(n2>=n1 && n2>=n3){

cout<<n2<<" is the maximum";}

else{

cout<<n3<<" is the maximum";}

return 0;

}

Explanation:

The program is written in C++ and to write this program, I assumed the three variables are integers. You can change from integer to double or float, if you wish.

This line declares n1, n2 and n3 as integers

int n1, n2, n3;

This line prompts user for three numbers

cout<<"Enter any three numbers: ";

This line gets user input for the three numbers

cin>>n1>>n2>>n3;

This if condition checks if n1 is the maximum and prints n1 as the maximum, if true

<em>if(n1>=n2 && n1>=n3){</em>

<em>cout<<n1<<" is the maximum";}</em>

This else if condition checks if n2 is the maximum and prints n2 as the maximum, if true

<em>else if(n2>=n1 && n2>=n3){</em>

<em>cout<<n2<<" is the maximum";}</em>

If the above conditions are false, then n3 is the maximum and this condition prints n3 as the maximum

<em>else{</em>

<em>cout<<n3<<" is the maximum";}</em>

return 0;

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3 years ago
In computer security what do the rows and columns correspond to in an 'Access Control Matrix'. What does each cell in the matrix
Tcecarenko [31]

Answer:

Explanation:

An Access Control Matrix ACM can be defined as a table that maps the permissions of a set of subjects to act upon a set of objects within a system. The matrix is a two-dimensional table with subjects down the columns and objects across the rows. The permissions of the subject to act upon a particular object are found in the cell that maps the subject to that object.

Summary

The rows correspond to the subject

The columns correspond to the object

What does each cell in the matrix contain? Answer: Each cell is the set of access rights for that subject to that object.

4 0
3 years ago
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