Evaluate each determinant.

Verified step by step guidance

1. Identify the size of the determinant matrix (e.g., 2x2 or 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 for a 3x3 matrix, expand along the first row: \(\det = a \times \det(M_{11}) - b \times \det(M_{12}) + c \times \det(M_{13})\), where \(M_{ij}\) are the 2x2 minors obtained by removing the \(i\)-th row and \(j\)-th column.
5. Calculate each 2x2 minor determinant using the 2x2 formula, then combine these results according to the cofactor expansion to find the determinant of the original matrix.

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.

Methods for Calculating Determinants

Determinants can be calculated using various methods depending on the 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 process. 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.