Question

Homework 4: please use python to find the answers. 1. Let A= 1 2 3 4 4 6 7 8 9 (a) What is A(2,3) (b) What is AT ? (c) Does A-1 exist? If yes, what is the value? 2. Let 2x + 3y + 4z = 1 -y +2 = 2 4x + y - 2 3 (a) What is the coefficient matrix? (b) Rewrite the linear system in the matrix form AX = b where X = (0) (c) Solve the equation by X = A-16 - use python command:

Answer 1

### NOTE - EXECUTE THE BOLD COMMANDS T GET THE OUTPUT BELOW THEM ###

import numpy as np

In [3]:

m = [[1,2,3],[4,4,6],[7,8,9]]

#declaring the matrix

In [7]:

​

A = np.array(m)

print('The A(2,3) is : '+str(A[1][2]))

#printing the element at 2,3 location

The m(2,3) is : 6

In [10]:

arr1_transpose = A.transpose()

​

print(f'Transposed Array:\n{arr1_transpose}')

Transposed Array:
[[1 4 7]
 [2 4 8]
 [3 6 9]]

In [12]:

DetA = np.linalg.det(A)

print("The determinant of A is : ",DetA)

The determinant of A is :  12.0

In [13]:

#since DetA>0 so inverse exist 

print("The inverse of A is : ",np.linalg.inv(A))

The inverse of A is :  [[-1.00000000e+00  5.00000000e-01  1.26882631e-16]
 [ 5.00000000e-01 -1.00000000e+00  5.00000000e-01]
 [ 3.33333333e-01  5.00000000e-01 -3.33333333e-01]]

INTENDATION -

In [1] : import numpy as np In [3]: m = [[1,2,3], [4,4,6],[7,8,9]] #declaring the matrix In [7]: A = np.array(m) print('The A(2,3) is : '+str(A[1][2])) #printing the element at 2,3 Location The m(2,3) is : 6 In (10): arri_transpose = A. transpose() print(f'Transposed Array: \n{arri_transpose}') Transposed Array: [(1 4 7] [248] (3 6 9]] In [12]: DetA = np. linalg.det(A) print("The determinant of A is : ",Deta) The determinant of A is : 12.8 In [13]: #since Detare so inverse exist print("The inverse of A is : ",np.linalg.inv(A)) The inverse of Ais : [[-1.888888eee+80 5.900000002-01 1.268826312-16] [ 5.00e0e8e8e-e1 -1.eeeeeeeee-80 5.88eeeeeee-e1] [ 3.33333333e-01 5. e8800002e-01 -3.33333333e-21]] In [ ]:

QUESTION 2 -

part 1 and 3

import numpy as np

In [2]:

A = [[2,3,4],[1,-1,1],[4,1,-1]]

A = np.array(A)

In [4]:

print("The coeffecient matrix is : \n",A)

#printig the coeffecient matrix

The coeffecient matrix is : 
 [[ 2  3  4]
 [ 1 -1  1]
 [ 4  1 -1]]

In [6]:

B = np.array([1, 2, 3])

X = np.linalg.inv(A).dot(B)

#SOLUUTION  BY X = A^-1 B

​

print("The x y z values in array form are: \n",X)

The x y z values in array form are: 
 [ 1.         -0.71428571  0.28571429]

solution to part 2-

2 *8221 [3 z o Y. ] y Z 3 AX : B 2 3 4 x (3] 2 3 -) st 4 -)

output -

In [1]: import numpy as np In [2]: A = [[2,3,4],[1,-1,1],[4,1,-1]] A = np.array(A) In [4]: print("The coeffecient matrix is : \n", A) #printig the coeffecient matrix The coeffecient matrix is : [[ 2 3 4] [ 1-1 1] [ 4 1 -1]] In [6]: B = np.array([1, 2, 3]) X = np.linalg.inv(A).dot(B) #SOLUUTION BY X = A-1 B print("The x y z values in array form are: \n",x) The x y z values in array form are: [ 1. -0.71428571 0.28571429]

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