18.3: Using Cramer’s rule to solve Ax=b

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Let \(Ax=b\) be a system of \(n\) linear equations in \(n\) variables such that \(|A| \neq 0\). The system has a unique solution given by:

\[x_1 = \frac{|A_1|}{|A|}, x_2 = \frac{|A_2|}{|A|}, \ldots, x_n = \frac{|A_n|}{|A|} \nonumber \]

where \(A_i\) is the matrix obtained by replacing column \(i\) of \(A\) with \(b\). The following function generates \(A_i\) by replacing the \(i\)th column of \(A\) with \(b\):

def makeAi(A,i,b):
    '''Replace the ith column in A with b'''
    if type(A) == np.matrix:
        A = A.tolist()
    if type(b) == np.matrix:
        b = b.tolist()
    Ai = copy.deepcopy(A)
    for j in range(len(Ai)):
        Ai[j][i] = b[j][0]
    return Ai

Do This

Create a new function called cramersRule, which takes \(A\) and \(b\) and returns \(x\) using the Cramer’s rule.

Note: Use numpy and NOT mydet to find the required determinants. mydet is too slow.

# Stub code. Replace the np.linalg.solve code with your answer

def cramersRule(A,b):
    detA = np.linalg.det(A)
    x = []    
    #####Start of your code here#####  
 

    #####End of your code here#####  
    
    return x

Question

Test your cramersRule function on the following system of linear equations:

\[ x_1 + 2x_2 = 3 \nonumber \]

\[3x_1 + x_2 = -1 \nonumber \]

#Put your answer to the above question here

Question

Verify the above answer by using the np.linalg.solve function:

#Put your answer to the above question here

Question

Test your cramersRule function on the following system of linear equations and verify the answer by using the np.linalg.solve function:

\[ x_1 + 2x_2 +x_3 = 9 \nonumber \]

\[ x_1 + 3x_2 - x_3 = 4 \nonumber \]

\[ x_1 + 4x_2 - x_3 = 7 \nonumber \]

#Put your answer to the above question here

Question

Cramer’s rule is a \(O(n!)\) algorithm and the Gauss-Jordan (or Gaussian) elimination is \(O(n^3)\). What advantages does Cramer’s rule have over elimination?