Determinants are really important when we try to solve systems of linear equations. They help us understand when there is a unique solution.

When we write a system of equations, we can use a matrix form, which looks like this: ( A\mathbf{x} = \mathbf{b} ). Here, ( A ) is called the coefficient matrix. The determinant of this matrix is shown as ( \text{det}(A) ). The determinant tells us a lot about the solutions we might find.

1. Do Solutions Exist?

   • If ( \text{det}(A) \neq 0 ), that means there is one unique solution. This means that the rows (or columns) of the matrix ( A ) are different enough not to depend on each other. In simpler terms, the equations intersect at just one point.

   • If ( \text{det}(A) = 0 ), that means there is either no solution or multiple solutions. This happens when the rows (or columns) of the matrix are connected or related, meaning they describe the same line or do not touch at all.

2. What Kind of Solutions Do We Have?

   • When the determinant is zero, we could end up with either no solutions or infinitely many solutions. If there are infinitely many solutions, we can often write them using what we call free variables. This means that we can express the solutions in different ways, showing just how important determinants are for understanding the types of solutions we have.

3. Using Cramer's Rule:

   • Determinants are also used in something called Cramer's Rule. This rule gives us a way to find the unique solution for a system of linear equations. Using Cramer’s Rule, we solve each variable ( \mathbf{x}_i ) like this:
   $$x_i = \frac{\text{det}(A_i)}{\text{det}(A)},$$

   In this formula, ( A_i ) is the matrix we get when we replace the ( i^{th} ) column of ( A ) with the vector ( \mathbf{b} ). This helps us see how determined our solutions are.

4. Transformation and Stability:

   • Determinants also help us understand how stable our solutions are when we change things. When we swap rows, scale, or combine them, the determinant shows us how those changes affect the system.

In short, determinants are an essential part of linear algebra. They give us important clues about whether solutions exist, if they are unique, and what kind of solutions we can find in systems of linear equations.