Use the determinant theorems to evaluate each determinant. See Example 4.
$\begin{vmatrix}-4 & 1 & 4\\2& 0 & 1\\ 0&2&4\end{vmatrix}$

Verified step by step guidance

1. Identify the size and structure of the given determinant matrix (e.g., 2x2, 3x3, etc.) to determine which determinant theorems and properties apply.
2. Recall key determinant theorems such as: the determinant of a triangular matrix is the product of its diagonal entries, swapping two rows changes the sign of the determinant, multiplying a row by a scalar multiplies the determinant by that scalar, and the determinant of a product of matrices equals the product of their determinants.
3. Apply row operations or factor out constants from rows or columns where possible, using the determinant theorems to simplify the determinant calculation without expanding directly.
4. If the matrix is larger than 2x2 and no simplifications apply, use cofactor expansion along a row or column with the most zeros to reduce the determinant to smaller matrices.
5. Combine all the results from the above steps to write the determinant as a product or sum of simpler determinants or values, preparing it for final evaluation.

Key Concepts

Determinant of a Matrix

The determinant is a scalar value computed from a square matrix that provides important properties such as invertibility. It can be calculated using various methods, including expansion by minors or row operations, and is essential for solving systems of linear equations and understanding matrix behavior.

Determinant Theorems

Determinant theorems are rules that simplify the calculation of determinants, such as the effect of row swaps, scalar multiplication of rows, and adding multiples of one row to another. These theorems help reduce complex determinants into simpler forms without changing their values or by adjusting them predictably.

Row Operations and Their Impact on Determinants

Certain row operations affect the determinant in specific ways: swapping two rows changes the sign, multiplying a row by a scalar multiplies the determinant by that scalar, and adding a multiple of one row to another does not change the determinant. Understanding these effects is crucial for using determinant theorems effectively.