Calculate the Determinant of a 3x3 Matrix

Calculate the determinant of the 3x3 matrix: $\begin{pmatrix} 5 & 7 & 8 \\ 4 & -3 & 6 \\ 1 & 7 & 9 \end{pmatrix}$.

To calculate the determinant of a 3x3 matrix, such as the one given, you will employ the rule of Sarrus or cofactor expansion, which are foundational techniques in linear algebra. Understanding how to calculate a determinant is crucial because determinants provide valuable insights into the properties of a matrix. For instance, a non-zero determinant can indicate that a matrix is invertible or that a system of linear equations has a unique solution.

The process of finding the determinant involves a combination of multiplication and addition operations. You will take the products of diagonals in specific patterns and combine them algebraically. While the formula and computation may seem mechanical, it's important to understand that the determinant itself can tell you about the linear transformation represented by the matrix, such as whether it preserves area or volume in a transformed space.

Engaging with this problem not only helps reinforce computational skills but also enhances your conceptual understanding of how matrices function within vector spaces. By practicing determinant calculation, you are building a deeper intuition around matrix behavior in geometric transformations and complex algebraic structures.