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A 3x3 matrix is a square matrix with three rows and three columns. Each element in this matrix is identified by its position in the matrix, usually denoted by a subscript notation like a_{ij}. Here, 'i' represents the row number and 'j' represents the column number.

For example, in the given matrix:\[ \begin{array}{rrr} -1 & 2 & 4 \ 4 & -8 & -16 \ 0 & 0 & 0 \end{array} \],
\(-1\) is in the first row and first column (a_{11}), \(2\) is in the first row and second column (a_{12}), and so on.

The structure of a 3x3 matrix makes it possible to perform various mathematical operations, including calculating the determinant which is essential in linear algebra.

Determinant theorems provide rules that simplify the process of calculating determinants. These theorems help us understand and find the determinant quickly without tedious calculations.

Some key determinant theorems include:
• If any row or column of a matrix is made up entirely of zeros, the determinant of the matrix is zero.
• Swapping two rows or two columns of a matrix changes the sign of the determinant.
• The determinant of a triangular matrix (either upper or lower) is the product of its diagonal elements.

In the given exercise, we notice that the third row consists entirely of zeros. According to the first theorem mentioned, this means the determinant of the matrix is zero, simplifying our calculation significantly.

Understanding the properties of determinants can help greatly when solving for the determinant of matrices. Here are some core properties:
• The determinant of a matrix changes sign if two rows or two columns are swapped.
• Multiplying a row or a column of a matrix by a scalar multiplies the determinant by that scalar.
• Adding a multiple of a row (or column) to another row (or column) does not change the determinant.
• The determinant of an identity matrix is always 1.

These properties can often be combined to simplify the calculation of the determinant. For example, in the given matrix:
\[ \begin{array}{rrr} -1 & 2 & 4 \ 4 & -8 & -16 \ 0 & 0 & 0 \end{array} \]
, we use the property that a row or column of all zeros results in a determinant of zero. This property saved us from extensive calculations and provided the solution effortlessly.