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.). The method depends on the matrix size.
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 for a 3x3 matrix, pick a row or column, multiply each element by its corresponding cofactor, and sum the results: \(\det = a(ei - fh) - b(di - fg) + c(dh - eg)\).
5. After setting up the determinant expression, simplify the arithmetic step-by-step 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. It is calculated using specific rules depending on the matrix size, like the product of diagonal elements for 2x2 matrices or expansion by minors for larger matrices.

Properties of Determinants

Determinants have properties that simplify calculations, such as the determinant of a product equals the product of determinants, and swapping two rows changes the sign of the determinant. Understanding these properties helps in efficiently evaluating determinants without full expansion.

Methods for Evaluating Determinants

Common methods include expansion by minors (cofactor expansion), row reduction to upper triangular form, and leveraging special matrix types (like diagonal or triangular matrices). Choosing the right method depends on matrix size and structure to simplify the evaluation process.