In mathematics, systems of linear equations are a common problem that arises in various fields, such as physics, engineering, and economics. Matrices provide an efficient and elegant way to solve these systems, offering a powerful tool to handle complex scenarios.

To solve a system of equations using matrices, we first express the system in matrix form, known as an augmented matrix. Let's consider an example:

To create the augmented matrix, we assign the coefficients of the variables to a matrix and the constant terms to a vector column. In this case, the augmented matrix would be:

Next, we perform elementary row operations on the augmented matrix to simplify it. The goal is to transform the matrix into its reduced row-echelon form, an equivalent matrix where leading entries in each row are ones and there are only zeros above and below each leading entry.

Once we have the reduced row-echelon form, we can easily read the solution of the system. The values in the rightmost column of the matrix correspond to the solutions of the variables. In our example, the solution would be: x = 2 and y = 1.

Using matrices and determinants allows us to solve systems of equations efficiently, providing a systematic approach that can be scaled to larger systems and more complex scenarios. Now, let's try some practice problems to strengthen our understanding!