In linear algebra, determinants are very important when figuring out the solutions for linear systems. A determinant can tell us if a system has a solution, and if it does, whether that solution is one unique answer or many answers.

1. Invertibility and Consistency:

   • If the determinant of a matrix, called $A$, is not zero ($\det(A) \neq 0$), then the matrix can be inverted. This means the system has one unique solution, which we can find using $A^{-1} \mathbf{b}$.
   • On the flip side, if $\det(A) = 0$, the matrix is singular. This means there might be no solutions or there might be an endless number of solutions, depending on how $A$ relates to another matrix called the augmented matrix $[A|\mathbf{b}]$.

2. Geometric Interpretation:

   • When the determinant is not zero, it means the rows or columns of matrix $A$ are independent. This can be thought of as having a proper space where we can find one point where lines meet in a graph.
   • However, if the determinant is zero, it indicates dependence among the rows or columns. This can look like multiple planes in space that are either parallel (which means no solutions) or planes that lay on top of each other (which means there are infinitely many solutions).

So, understanding determinants is essential for looking at linear systems. They help us find out if solutions exist, and they also show us if those solutions are unique or countless.