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 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 value: \(\det = a_{ij}C_{ij} + a_{ik}C_{ik} + \dots\) where \(C_{ij}\) are cofactors.

Key Concepts

Determinant of a Matrix

The determinant is a scalar value computed from a square matrix that provides important properties 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.

Methods for Calculating Determinants

Determinants can be found using various methods depending on matrix size, including expansion by minors, cofactor expansion, and row reduction. For larger matrices, breaking down into smaller matrices or using properties like linearity simplifies the calculation. Understanding these methods is essential for efficient evaluation.

Properties of Determinants

Determinants have key properties such as changing sign when two rows are swapped, being zero if rows are linearly dependent, and multiplicative behavior under matrix multiplication. These properties aid in simplifying calculations and interpreting the determinant's meaning in algebraic contexts.