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: \(\det = \sum (-1)^{i+j} a_{ij} M_{ij}\), where \(M_{ij}\) is the minor determinant after removing row \(i\) and column \(j\).
5. Compute each minor determinant recursively if needed, then combine all terms according to the chosen method to find 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.

Methods for Calculating Determinants

Determinants can be calculated using various methods depending on the matrix size. For 2x2 and 3x3 matrices, formulas and expansion by minors or cofactors are common. Larger matrices often require row reduction or Laplace expansion. Understanding these methods is essential for evaluating determinants efficiently.

Properties of Determinants

Determinants have key properties such as linearity, the effect of row operations, and multiplicative behavior. For example, swapping two rows changes the sign of the determinant, and a matrix with a row of zeros has a determinant of zero. These properties simplify calculations and help verify results.