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The determinant of a matrix is a special number calculated from a square matrix. In the context of solving systems of linear equations using Cramer's Rule, the determinant plays a crucial role. When you have a matrix:\[A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\]the determinant is found using the formula:\[\text{det}(A) = ad - bc\]This formula gives a scalar value that can indicate various properties of the matrix, such as whether it is invertible. For example, in the exercise, the coefficient matrix had a determinant of 30. Since this is not zero, Cramer's Rule is applicable, meaning the system of equations has a unique solution. The determinant essentially "scales" how much the vector space defined by the matrix will be transformed, which has computational importance when solving linear equations.

A system of linear equations consists of multiple linear equations that share the same set of variables. It can visually be thought of as several lines in a plane, as in 2D, where each equation represents a line. The solution to the system is the point or points where these lines intersect. In our exercise, the system:

• 10x - 17y = 21
• 20x - 31y = 39

is 2-dimensional because we have two variables, x and y. There is a point (x, y) in the coordinate plane that satisfies both equations simultaneously. Solving such systems can be done using methods like substitution, elimination, or (as in our case) Cramer's Rule, which uses determinants to find a unique intersection point if the system is consistent and the coefficient matrix's determinant is non-zero.

The coefficient matrix in a system of linear equations is a matrix derived from the coefficients of the variables in the equations. It serves as a foundational tool in using Cramer's Rule. For the example given:

• From the equation 10x - 17y = 21
• From the equation 20x - 31y = 39

The coefficient matrix A is:\[A = \begin{pmatrix} 10 & -17 \ 20 & -31 \end{pmatrix}\]This matrix contains only the coefficients of x and y, excluding the constants. It organizes the system compactly and allows us to apply matrix operations, such as calculating the determinant or transforming it to solve for the variables using different mathematical techniques, including Cramer's Rule. A non-zero determinant of this matrix implies that the system has a unique solution, which was confirmed by solving the equations in this manner.