91Ó°ÊÓ

When we talk about a system of linear equations, we refer to a collection of linear equations that we solve together to find common solutions for the variables involved. These systems are fundamental in algebra and are the basis for more complex subjects.
In our exercise, we are dealing with a system of three linear equations with three variables. The focus is to determine whether there is a unique solution, which means that there is only one set of values for the variables \(x_1\text{, }x_2\text{, and }x_3\) that satisfies all three equations simultaneously. A unique solution is ideal because it gives a precise answer to the problem at hand. To establish the existence of a unique solution, we turn to the coefficient matrix and its determinant.

The coefficients of a linear equation system can be arranged into an array called the coefficient matrix. Each row of the matrix corresponds to one equation, and each column represents one variable.
In our example, the coefficient matrix \( A \) corresponds to the left-hand side of the system of equations, excluding the constants on the right. The concept of the coefficient matrix is crucial because it allows us to represent the system in a compact form and perform algebraic operations like determinant calculation, which are essential to analyze the properties of the system, such as whether the solutions are unique.

The determinant of a matrix is a scalar value that can tell us a lot about the matrix and the system it represents. Determinant calculation is particularly important because it helps in determining whether a system of equations has a unique solution, no solution, or infinitely many solutions.
For a 3x3 matrix, the determinant is calculated using a specific formula involving the multiplication and subtraction of certain elements of the array. The pattern follows a cross-multiplication for each element of the first row, along with alternating signs. The total value helps us understand the nature of the solutions to the system.

The uniqueness of the solution to a system of linear equations is directly linked to the determinant of the coefficient matrix. When the determinant is not zero, it indicates that the matrix is invertible, and there is a unique solution to the system.
The determinant being zero would suggest that the system has either no solutions or an infinite number of solutions, also known as being 'inconsistent' or 'dependent', respectively. Uniqueness is a favorable outcome in many mathematical and practical applications because it offers a single, unambiguous result.

Let's dive into the specifics for our 3x3 matrix from the exercise. The determination of the determinant for a 3x3 matrix involves a method that is a straightforward application of the formula:
calculation formula here(see original exercise and solution)
For the given matrix \( A \) from our exercise, the calculation yields a determinant of 6, which is a non-zero value, thus confirming the presence of a unique solution. The calculation is relatively simple, but it's important always to keep track of the signs and order of operations to avoid mistakes.