Evaluate each determinant.

Verified step by step guidance

1. Identify the size of the determinant matrix (e.g., 2x2, 3x3) to determine the appropriate method for evaluation.
2. For a 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\), use the formula for the determinant: \(\det = ad - bc\).
3. For a 3x3 matrix \(\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\), apply the rule of Sarrus or cofactor expansion to find the determinant.
4. If using cofactor expansion, select a row or column (usually one with zeros for simplicity), then calculate the sum of each element multiplied by its cofactor: \(\det = \sum a_{ij} C_{ij}\), where \(C_{ij} = (-1)^{i+j} M_{ij}\) and \(M_{ij}\) is the minor determinant.
5. Compute each minor determinant by removing the row and column of the element, then multiply by the corresponding sign and element, and finally sum all these products to get the determinant value.

Key Concepts

Determinant of a Matrix

The determinant is a scalar value computed from a square matrix that provides important properties about the matrix, such as invertibility. For a 2x2 matrix, it is calculated as ad - bc, where a, b, c, and d are the elements of the matrix. Determinants help in solving systems of linear equations and understanding matrix behavior.

Properties of Determinants

Determinants have specific properties, such as changing sign when two rows are swapped, being zero if rows are linearly dependent, and the determinant of a product equals the product of determinants. These properties simplify calculations and help verify results when evaluating determinants.

Methods for Evaluating Determinants

Determinants can be evaluated using various methods depending on matrix size, including expansion by minors and cofactors for larger matrices, and direct formulas for 2x2 or 3x3 matrices. Understanding these methods allows efficient and accurate computation of determinants.