Question

Use the determinant theorems to evaluate each determinant. See Example 4.﻿∣−414201024∣\begin{vmatrix}

-4 & 1 & 4\

2& 0 & 1\ 0&2&4

\end{vmatrix}​−420​102​414​​

Accepted Answer

Identify the size and structure of the given determinant matrix (e.g., 2x2, 3x3, etc.) to determine which determinant theorems and properties apply. 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. 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. 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. 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.

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

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Key Concepts

Determinant of a Matrix

Determinant Theorems

Row Operations and Their Impact on Determinants

Use the determinant theorems to evaluate each determinant. See Example 4.∣−414201024∣\(\begin{vmatrix}\)-4 & 1 & 4\\2& 0 & 1\\ 0&2&4\(\end{vmatrix}\)−420102414