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Direct Methods: Cramer’s Rule

The Method

Cramer’s rule is one of the early methods to solve systems of linear equations $Ax=b$ . It is dated back to the eighteenth century! The method works really well; however, it is inefficient when the number of equations is very large. Nevertheless, with the current computing capabilities, the method can still be pretty fast. The method works using the following steps. First, the determinant of the matrix $A$ is calculated. Each component $x_i$ of the unknowns can be computed as:

$x_i=\frac{\det{D_i}}{\det{A}}$

where $D_i$ is the matrix formed by replacing the $i^{th}$ column in $A$ with the vector $b$ .

Example 1

Consider the linear system of equations $Ax=b$ defined as:

$\left(\begin{matrix}1&2\\-2&1\end{matrix}\right)\left(\begin{array}{c}x_1\\x_2\end{array}\right)=\left(\begin{array}{c}-1\\2\end{array}\right)$

$\det{A}=1\times 1 + 2\times 2=5$ . We will form the following two matrices by replacing the first column in $A$ with $b$ and then the second column in $A$ with $b$ :

$D_1=\left(\begin{matrix}-1&2\\2&1\end{matrix}\right) \qquad D_2=\left(\begin{matrix}1&-1\\-2&2\end{matrix}\right)$

The determinants of these matrices are:

$\det{D_1}=-1\times 1-2\times 2=-5 \qquad \det{D_2}=1\times 2-2\times 1=0$

Therefore:

$x_1=\frac{\det{D_1}}{\det{A}}=\frac{-5}{5}=-1 \qquad x_2=\frac{\det{D_2}}{\det{A}}=\frac{0}{5}=0$

The following is code that can be used for Cramer’s rule for a two-dimensional system of equations.

View Mathematica Code

A = {{a11, a12}, {a21, a22}};
AT = Transpose[A]
b = {b1, b2};
s1 = Drop[AT, {1}]
s2 = Drop[AT, {2}]
D1T = Insert[s1, b, 1]
D2T = Insert[s2, b, 2]
D1 = Transpose[D1T]
D2 = Transpose[D2T]
x1 = Det[D1]/Det[A]
x2 = Det[D2]/Det[A]

View Python Code

import sympy as sp
a11, a12, a21, a22 = sp.symbols("a11 a12 a21 a22")
A = sp.Matrix([[a11, a12], [a21, a22]])
print("A:",A)
AT = A.T
print("A Transpose:",AT)
s1 = sp.Matrix(AT)
s1.row_del(0)
print("s1:",s1)
s2 = sp.Matrix(AT)
s2.row_del(1)
print("s2:",s2)
b1, b2 = sp.symbols("b1 b2")
b = sp.Matrix([[b1, b2]])
D1T = sp.Matrix(s1)
D1T = D1T.row_insert(0,b)
print("D1T:",D1T)
D2T = sp.Matrix(s2)
D2T = D2T.row_insert(1,b)
print("D2T:",D2T)
D1 = D1T.T
print("D1T Transpose:",D1)
D2 = D2T.T
print("D2T Transpose:",D2)
x1 = D1.det()/A.det()
print("x1:",x1)
x2 = D2.det()/A.det()
print("x2:",x2)

Example 2

Consider the linear system of equations $Ax=b$ defined as:

$\left(\begin{matrix}1&2&3\\2&1&4\\1&3&5\end{matrix}\right)\left(\begin{array}{c}x_1\\x_2\\x_3\end{array}\right)=\left(\begin{array}{c}-4\\8\\0\end{array}\right)$

$\det{A}=-4$ . We will form the following three matrices by replacing the corresponding columns in $A$ with $b$ :

$D_1=\left(\begin{matrix}-4&2&3\\8&1&4\\0&3&5\end{matrix}\right) \qquad D_2=\left(\begin{matrix}1&-4&3\\2&8&4\\1&0&5\end{matrix}\right) \qquad D_3=\left(\begin{matrix}1&2&-4\\2&1&8\\1&3&0\end{matrix}\right)$

The determinants of these matrices are:

$\det{D_1}=20 \qquad \det{D_2}=40 \qquad \det{D_3}=-28$

Therefore:

$x_1=\frac{\det{D_1}}{\det{A}}=-5 \qquad x_2=\frac{\det{D_2}}{\det{A}}=-10 \qquad x_3=\frac{\det{D_3}}{\det{A}}=7$

The following procedure applies the Cramer’s rule to a general system. Test the procedure to see how fast it is for higher dimensions.

View Mathematica Code

Cramer[A_, b_] := (n = Length[A];
  Di = Table[A, {i, 1, n}];
  Do[Di[[i]][[All, i]] = b, {i, 1, n}];
  x = Table[Det[Di[[i]]]/Det[A], {i, 1, n}])

A = {{1, 2, 3, 5}, {2, 0, 1, 4}, {1, 2, 2, 5}, {4, 3, 2, 2}};
b = {-4, 8, 0, 10};
Cramer[A, b]

View Python Code

import numpy as np
def Cramer(A, b):
  n = len(A)
  Di = [np.array(A) for i in range(n)]
  for i in range(n):
    Di[i][:,i] = b
  return [np.linalg.det(Di[i])/np.linalg.det(A) for i in range(n)]
A = [[1, 2, 3, 5], [2, 0, 1, 4], [1, 2, 2, 5], [4, 3, 2, 2]]
b = [-4, 8, 0, 10]
Cramer(A, b)

The following link provides the MATLAB code for implementing the Cramer’s rule.

MATLAB file Download

Open Educational Resources

Direct Methods: Cramer’s Rule

The Method

Example 1

Example 2

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