Solving Systems Of Linear Equations Using Cramer's Rule

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In mathematics, solving systems of linear equations is a fundamental concept with numerous applications across various fields. One powerful method for solving such systems is Cramer's Rule, a formula that provides solutions in terms of determinants. This article explores Cramer's Rule and demonstrates its application through several examples.

Understanding Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. It is particularly useful when dealing with systems that have the same number of equations as variables and a unique solution. The method involves calculating several determinants: the determinant of the coefficient matrix and determinants formed by replacing columns of the coefficient matrix with the constant terms.

To apply Cramer's Rule, consider a system of n linear equations with n variables:

a₁₁x₁ + a₁₂x₂ + ... + a₁nxn = b₁
a₂₁x₁ + a₂₂x₂ + ... + a₂nxn = b₂
...
an₁x₁ + an₂x₂ + ... + annxn = bn

This system can be represented in matrix form as Ax = b, where A is the coefficient matrix, x is the column vector of variables, and b is the column vector of constants.

The determinant of the coefficient matrix, denoted as |A|, plays a crucial role in Cramer's Rule. If |A| is non-zero, the system has a unique solution, and Cramer's Rule can be applied. If |A| is zero, the system either has no solution or infinitely many solutions, and Cramer's Rule cannot be used.

To find the solution for a variable xᵢ, we calculate a new determinant, |Aᵢ|, by replacing the i-th column of A with the column vector b. The solution for xᵢ is then given by:

xᵢ = |Aᵢ| / |A|

This process is repeated for each variable to obtain the complete solution of the system.

Advantages and Limitations of Cramer's Rule

Cramer's Rule offers several advantages:

  • Direct Solution: It provides a direct formula for finding the solution, without the need for iterative methods.
  • Conceptual Clarity: The method is conceptually straightforward and easy to understand.
  • Applicability: It is particularly useful for small systems of equations and for systems where only a few variables need to be determined.

However, Cramer's Rule also has limitations:

  • Computational Cost: Calculating determinants can be computationally expensive for large systems, making other methods like Gaussian elimination more efficient.
  • Singular Systems: Cramer's Rule cannot be applied to systems where the determinant of the coefficient matrix is zero.
  • Numerical Instability: For ill-conditioned systems, where small changes in the coefficients lead to large changes in the solution, Cramer's Rule can be numerically unstable.

Solving Systems of Linear Equations: Examples with Detailed Explanations

Let's delve into solving specific systems of linear equations using Cramer's Rule, providing step-by-step solutions and detailed explanations.

Example A

Solve the following system of linear equations using Cramer's Rule:

x - 2y - z = -7
2x + y + z = 0
3x - 5y + 8z = 13

  1. Form the coefficient matrix (A) and the constant vector (b):

    A = | 1  -2  -1 |
    | 2   1   1 |
    | 3  -5   8 |

b = | -7 |
    |  0 |
    | 13 |

  2. Calculate the determinant of the coefficient matrix (|A|):

    |A| = 1 * (1*8 - 1*(-5)) - (-2) * (2*8 - 1*3) + (-1) * (2*(-5) - 1*3)
    = 1 * (8 + 5) + 2 * (16 - 3) - 1 * (-10 - 3)
    = 1 * 13 + 2 * 13 - 1 * (-13)
    = 13 + 26 + 13
    = 52

    Since |A| = 52 ≠ 0, the system has a unique solution, and we can proceed with Cramer's Rule.

  3. Calculate the determinant for x (|Ax|): Replace the first column of A with b.

    Ax = | -7  -2  -1 |
     |  0   1   1 |
     | 13  -5   8 |

|Ax| = -7 * (1*8 - 1*(-5)) - (-2) * (0*8 - 1*13) + (-1) * (0*(-5) - 1*13)
     = -7 * (8 + 5) + 2 * (0 - 13) - 1 * (0 - 13)
     = -7 * 13 + 2 * (-13) - 1 * (-13)
     = -91 - 26 + 13
     = -104

  4. Calculate the determinant for y (|Ay|): Replace the second column of A with b.

    Ay = | 1  -7  -1 |
     | 2   0   1 |
     | 3  13   8 |

|Ay| = 1 * (0*8 - 1*13) - (-7) * (2*8 - 1*3) + (-1) * (2*13 - 0*3)
     = 1 * (0 - 13) + 7 * (16 - 3) - 1 * (26 - 0)
     = 1 * (-13) + 7 * 13 - 1 * 26
     = -13 + 91 - 26
     = 52

  5. Calculate the determinant for z (|Az|): Replace the third column of A with b.

    Az = | 1  -2  -7 |
     | 2   1   0 |
     | 3  -5  13 |

|Az| = 1 * (1*13 - 0*(-5)) - (-2) * (2*13 - 0*3) + (-7) * (2*(-5) - 1*3)
     = 1 * (13 - 0) + 2 * (26 - 0) - 7 * (-10 - 3)
     = 1 * 13 + 2 * 26 - 7 * (-13)
     = 13 + 52 + 91
     = 156

  6. Apply Cramer's Rule to find the solutions:

    x = |Ax| / |A| = -104 / 52 = -2
y = |Ay| / |A| = 52 / 52 = 1
z = |Az| / |A| = 156 / 52 = 3

    Therefore, the solution to the system is x = -2, y = 1, and z = 3.

Example B

Solve the following system of linear equations using Cramer's Rule:

x - y - z = -2
x + 4z = 6
y - 2z = 1

  1. Form the coefficient matrix (A) and the constant vector (b):

    A = | 1  -1  -1 |
    | 1   0   4 |
    | 0   1  -2 |

b = | -2 |
    |  6 |
    |  1 |

  2. Calculate the determinant of the coefficient matrix (|A|):

    |A| = 1 * (0*(-2) - 4*1) - (-1) * (1*(-2) - 4*0) + (-1) * (1*1 - 0*0)
    = 1 * (0 - 4) + 1 * (-2 - 0) - 1 * (1 - 0)
    = 1 * (-4) + 1 * (-2) - 1 * 1
    = -4 - 2 - 1
    = -7

    Since |A| = -7 ≠ 0, the system has a unique solution.

  3. Calculate the determinant for x (|Ax|):

    Ax = | -2  -1  -1 |
     |  6   0   4 |
     |  1   1  -2 |

|Ax| = -2 * (0*(-2) - 4*1) - (-1) * (6*(-2) - 4*1) + (-1) * (6*1 - 0*1)
     = -2 * (0 - 4) + 1 * (-12 - 4) - 1 * (6 - 0)
     = -2 * (-4) + 1 * (-16) - 1 * 6
     = 8 - 16 - 6
     = -14

  4. Calculate the determinant for y (|Ay|):

    Ay = | 1  -2  -1 |
     | 1   6   4 |
     | 0   1  -2 |

|Ay| = 1 * (6*(-2) - 4*1) - (-2) * (1*(-2) - 4*0) + (-1) * (1*1 - 6*0)
     = 1 * (-12 - 4) + 2 * (-2 - 0) - 1 * (1 - 0)
     = 1 * (-16) + 2 * (-2) - 1 * 1
     = -16 - 4 - 1
     = -21

  5. Calculate the determinant for z (|Az|):

    Az = | 1  -1  -2 |
     | 1   0   6 |
     | 0   1   1 |

|Az| = 1 * (0*1 - 6*1) - (-1) * (1*1 - 6*0) + (-2) * (1*1 - 0*0)
     = 1 * (0 - 6) + 1 * (1 - 0) - 2 * (1 - 0)
     = 1 * (-6) + 1 * 1 - 2 * 1
     = -6 + 1 - 2
     = -7

  6. Apply Cramer's Rule to find the solutions:

    x = |Ax| / |A| = -14 / -7 = 2
y = |Ay| / |A| = -21 / -7 = 3
z = |Az| / |A| = -7 / -7 = 1

    Therefore, the solution to the system is x = 2, y = 3, and z = 1.

Example C

Solve the following system of linear equations using Cramer's Rule:

x - 2y - 5z = -12
2x - y = 7
5y + 6z = 4x + 1

First, we need to rewrite the third equation to bring all variables to one side:

4x - 5y - 6z = -1

Now, the system of equations is:

x - 2y - 5z = -12
2x - y = 7
4x - 5y - 6z = -1

  1. Form the coefficient matrix (A) and the constant vector (b):

    A = | 1  -2  -5 |
    | 2  -1   0 |
    | 4  -5  -6 |

b = | -12 |
    |  7 |
    | -1 |

  2. Calculate the determinant of the coefficient matrix (|A|):

    |A| = 1 * ((-1)*(-6) - 0*(-5)) - (-2) * (2*(-6) - 0*4) + (-5) * (2*(-5) - (-1)*4)
    = 1 * (6 - 0) + 2 * (-12 - 0) - 5 * (-10 + 4)
    = 1 * 6 + 2 * (-12) - 5 * (-6)
    = 6 - 24 + 30
    = 12

    Since |A| = 12 ≠ 0, the system has a unique solution.

  3. Calculate the determinant for x (|Ax|):

    Ax = | -12  -2  -5 |
     |   7  -1   0 |
     |  -1  -5  -6 |

|Ax| = -12 * ((-1)*(-6) - 0*(-5)) - (-2) * (7*(-6) - 0*(-1)) + (-5) * (7*(-5) - (-1)*(-1))
     = -12 * (6 - 0) + 2 * (-42 - 0) - 5 * (-35 - 1)
     = -12 * 6 + 2 * (-42) - 5 * (-36)
     = -72 - 84 + 180
     = 24

  4. Calculate the determinant for y (|Ay|):

    Ay = | 1  -12  -5 |
     | 2    7   0 |
     | 4   -1  -6 |

|Ay| = 1 * (7*(-6) - 0*(-1)) - (-12) * (2*(-6) - 0*4) + (-5) * (2*(-1) - 7*4)
     = 1 * (-42 - 0) + 12 * (-12 - 0) - 5 * (-2 - 28)
     = 1 * (-42) + 12 * (-12) - 5 * (-30)
     = -42 - 144 + 150
     = -36

  5. Calculate the determinant for z (|Az|):

    Az = | 1  -2  -12 |
     | 2  -1    7 |
     | 4  -5   -1 |

|Az| = 1 * ((-1)*(-1) - 7*(-5)) - (-2) * (2*(-1) - 7*4) + (-12) * (2*(-5) - (-1)*4)
     = 1 * (1 + 35) + 2 * (-2 - 28) - 12 * (-10 + 4)
     = 1 * 36 + 2 * (-30) - 12 * (-6)
     = 36 - 60 + 72
     = 48

  6. Apply Cramer's Rule to find the solutions:

    x = |Ax| / |A| = 24 / 12 = 2
y = |Ay| / |A| = -36 / 12 = -3
z = |Az| / |A| = 48 / 12 = 4

    Therefore, the solution to the system is x = 2, y = -3, and z = 4.

Example D

Solve the following system of linear equations using Cramer's Rule:

x + y + z = 7
x - y + z = 3
x + y - z = 3

  1. Form the coefficient matrix (A) and the constant vector (b):

    A = | 1   1   1 |
    | 1  -1   1 |
    | 1   1  -1 |

b = | 7 |
    | 3 |
    | 3 |

  2. Calculate the determinant of the coefficient matrix (|A|):

    |A| = 1 * ((-1)*(-1) - 1*1) - 1 * (1*(-1) - 1*1) + 1 * (1*1 - (-1)*1)
    = 1 * (1 - 1) - 1 * (-1 - 1) + 1 * (1 + 1)
    = 1 * 0 - 1 * (-2) + 1 * 2
    = 0 + 2 + 2
    = 4

    Since |A| = 4 ≠ 0, the system has a unique solution.

  3. Calculate the determinant for x (|Ax|):

    Ax = | 7   1   1 |
     | 3  -1   1 |
     | 3   1  -1 |

|Ax| = 7 * ((-1)*(-1) - 1*1) - 1 * (3*(-1) - 1*3) + 1 * (3*1 - (-1)*3)
     = 7 * (1 - 1) - 1 * (-3 - 3) + 1 * (3 + 3)
     = 7 * 0 - 1 * (-6) + 1 * 6
     = 0 + 6 + 6
     = 12

  4. Calculate the determinant for y (|Ay|):

    Ay = | 1   7   1 |
     | 1   3   1 |
     | 1   3  -1 |

|Ay| = 1 * (3*(-1) - 1*3) - 7 * (1*(-1) - 1*1) + 1 * (1*3 - 3*1)
     = 1 * (-3 - 3) - 7 * (-1 - 1) + 1 * (3 - 3)
     = 1 * (-6) - 7 * (-2) + 1 * 0
     = -6 + 14 + 0
     = 8

  5. Calculate the determinant for z (|Az|):

    Az = | 1   1   7 |
     | 1  -1   3 |
     | 1   1   3 |

|Az| = 1 * ((-1)*3 - 3*1) - 1 * (1*3 - 3*1) + 7 * (1*1 - (-1)*1)
     = 1 * (-3 - 3) - 1 * (3 - 3) + 7 * (1 + 1)
     = 1 * (-6) - 1 * 0 + 7 * 2
     = -6 - 0 + 14
     = 8

  6. Apply Cramer's Rule to find the solutions:

    x = |Ax| / |A| = 12 / 4 = 3
y = |Ay| / |A| = 8 / 4 = 2
z = |Az| / |A| = 8 / 4 = 2

    Therefore, the solution to the system is x = 3, y = 2, and z = 2.

Conclusion

Cramer's Rule provides a systematic method for solving systems of linear equations using determinants. While it may not be the most efficient method for large systems, its conceptual clarity and direct formula make it a valuable tool in various mathematical and scientific applications. The examples discussed illustrate the step-by-step application of Cramer's Rule, showcasing its effectiveness in finding unique solutions to linear systems. Understanding both the advantages and limitations of Cramer's Rule allows for its appropriate use in problem-solving.

By mastering Cramer's Rule, students and professionals can confidently tackle linear systems and gain a deeper understanding of linear algebra principles. Whether in engineering, economics, or computer science, the ability to solve systems of linear equations is an invaluable skill, and Cramer's Rule provides a powerful approach to this fundamental task. Therefore, understanding and applying Cramer's Rule is a crucial step in developing problem-solving skills in various domains.