Determinant and Unique Solutions

Oct 13, 2025 by wbrum.org 57 views

Introduction

In linear algebra, a system of linear equations is a collection of equations in which the unknowns are related through linear combinations. The solution to such a system is a set of values for the unknowns that satisfy all the equations simultaneously. In this article, we will explore how to use the determinant of the coefficient matrix to determine whether a linear system has a unique solution.

What is a Determinant?

A determinant is a scalar value that can be calculated from the elements of a square matrix. It is denoted by the symbol |A| or det(A) and is used to determine the solvability of a system of linear equations. The determinant of a 2x2 matrix A is given by:

|A| = ad - bc

where a, b, c, and d are the elements of the matrix.

Determinant and Unique Solutions

The determinant of the coefficient matrix is used to determine whether a linear system has a unique solution. If the determinant is non-zero, then the system has a unique solution. On the other hand, if the determinant is zero, then the system has either no solution or infinitely many solutions.

Example 1: System of Two Equations

Consider the following system of two linear equations:

{ \begin{array}{l} x + 3y = 4 \\ 3x - y = 5 \end{array} \}

To determine whether this system has a unique solution, we need to calculate the determinant of the coefficient matrix.

Coefficient Matrix

The coefficient matrix of the system is:

{ \begin{array}{l} \begin{bmatrix} 1 & 3 \\ 3 & -1 \end{bmatrix} \end{array} \}

Determinant of the Coefficient Matrix

The determinant of the coefficient matrix is:

|A| = (1)(-1) - (3)(3) = -1 - 9 = -10

Unique Solution

Since the determinant is non-zero, the system has a unique solution.

Example 2: System of Three Equations

Consider the following system of three linear equations:

{ \begin{array}{l} x + 2y - z = 8 \\ x - y + 2z = 0 \end{array} \}

To determine whether this system has a unique solution, we need to calculate the determinant of the coefficient matrix.

Coefficient Matrix

The coefficient matrix of the system is:

{ \begin{array}{l} \begin{bmatrix} 1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix} \end{array} \}

Determinant of the Coefficient Matrix

The determinant of the coefficient matrix is:

|A| = (1)(-1) - (2)(1) = -1 - 2 = -3

Unique Solution

Since the determinant is non-zero, the system has a unique solution.

Conclusion

In conclusion, the determinant of the coefficient matrix is a useful tool for determining whether a linear system has a unique solution. If the determinant is non-zero, then the system has a unique solution. On the other hand, if the determinant is zero, then the system has either no solution or infinitely many solutions.

Future Work

In future work, we can explore other methods for determining unique solutions in linear systems, such as using the rank of the coefficient matrix or the null space of the matrix.

References

[1] Anton, H. (2018). Elementary Linear Algebra. John Wiley & Sons.
[2] Strang, G. (2016). Linear Algebra and Its Applications. Cengage Learning.

Glossary

Determinant: A scalar value that can be calculated from the elements of a square matrix.
Coefficient Matrix: A matrix whose elements are the coefficients of the variables in a system of linear equations.
Unique Solution: A solution to a system of linear equations that is the only possible solution.
No Solution: A system of linear equations that has no solution.
Infinitely Many Solutions: A system of linear equations that has an infinite number of solutions.

Introduction

In our previous article, we explored how to use the determinant of the coefficient matrix to determine whether a linear system has a unique solution. In this article, we will answer some frequently asked questions about determining unique solutions in linear systems.

Q: What is the significance of the determinant in determining unique solutions?

A: The determinant of the coefficient matrix is a scalar value that can be calculated from the elements of the matrix. If the determinant is non-zero, then the system has a unique solution. On the other hand, if the determinant is zero, then the system has either no solution or infinitely many solutions.

Q: How do I calculate the determinant of a 2x2 matrix?

A: The determinant of a 2x2 matrix A is given by:

|A| = ad - bc

where a, b, c, and d are the elements of the matrix.

Q: What if the determinant is zero? Does that mean the system has no solution?

A: Not necessarily. If the determinant is zero, then the system may have either no solution or infinitely many solutions. To determine which, you need to perform row operations on the augmented matrix to put it in reduced row echelon form.

Q: Can I use the determinant to determine the number of solutions to a system of linear equations?

A: Yes, you can use the determinant to determine the number of solutions to a system of linear equations. If the determinant is non-zero, then the system has a unique solution. If the determinant is zero, then the system has either no solution or infinitely many solutions.

Q: How do I determine the number of solutions to a system of linear equations using the determinant?

A: To determine the number of solutions to a system of linear equations using the determinant, follow these steps:

Calculate the determinant of the coefficient matrix.
If the determinant is non-zero, then the system has a unique solution.
If the determinant is zero, then perform row operations on the augmented matrix to put it in reduced row echelon form.
If the reduced row echelon form has a row of the form [0 0 | b], where b is a non-zero constant, then the system has no solution.
If the reduced row echelon form has a row of the form [0 0 | 0], then the system has infinitely many solutions.

Q: Can I use the determinant to determine the rank of a matrix?

A: Yes, you can use the determinant to determine the rank of a matrix. If the determinant is non-zero, then the rank of the matrix is equal to the number of rows or columns. If the determinant is zero, then the rank of the matrix is less than the number of rows or columns.

Q: How do I determine the rank of a matrix using the determinant?

A: To determine the rank of a matrix using the determinant, follow these steps:

Calculate the determinant of the matrix.
If the determinant is non-zero, then the rank of the matrix is equal to the number of rows or columns.
If the determinant is zero, then the rank of the matrix is less than the number of rows or columns.

Q: Can I use the determinant to determine the null space of a matrix?

A: Yes, you can use the determinant to determine the null space of a matrix. If the determinant is non-zero, then the null space of the matrix is the zero vector. If the determinant is zero, then the null space of the matrix is a non-zero vector.

Q: How do I determine the null space of a matrix using the determinant?

A: To determine the null space of a matrix using the determinant, follow these steps:

Calculate the determinant of the matrix.
If the determinant is non-zero, then the null space of the matrix is the zero vector.
If the determinant is zero, then the null space of the matrix is a non-zero vector.