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: \(\det = a \times d - b \times c\).
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: \(\det = \sum (-1)^{i+j} \times \text{element}_{ij} \times \det(\text{minor}_{ij})\).
5. Compute the minors (determinants of smaller matrices) as needed, then combine all terms according to the chosen method to express 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 several key properties: 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. These properties simplify determinant calculation and are useful in matrix manipulation.

Methods for Evaluating Determinants

Determinants can be evaluated using various methods depending on matrix size, such as expansion by minors and cofactors for larger matrices or row reduction to triangular form. Understanding these methods allows efficient calculation and deeper insight into matrix structure.