Evaluate each determinant.

Verified step by step guidance

1. Identify the size of the matrix for which you need to evaluate the determinant (e.g., 2x2, 3x3, etc.).
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 to simplify calculations), then calculate the minors and cofactors for each element in that row or column.
5. Sum the products of each element and its corresponding cofactor to get the determinant: \( \det = a_{ij}C_{ij} + a_{ik}C_{ik} + \ldots \), where \( C_{ij} \) are the cofactors.

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. It is calculated using specific rules depending on the matrix size, like expansion by minors or row operations.

Methods for Evaluating Determinants

Determinants can be evaluated using various methods including cofactor expansion, row reduction to echelon form, or special shortcuts for certain matrix types. Choosing an efficient method simplifies the calculation process.

Properties of Determinants

Determinants have properties such as linearity, the effect of row swaps (which change the sign), and the determinant of a product equals the product of determinants. Understanding these helps in simplifying and verifying determinant calculations.