Evaluate each determinant. See Example 1.

Verified step by step guidance

1. Identify the matrix from which the determinant needs to be calculated. The matrix should be a square matrix (same number of rows and columns).
2. If the matrix is a 2x2 matrix, use the formula \(ad - bc\) where \(a, b, c,\) and \(d\) are the elements of the matrix arranged as \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\).
3. For a 3x3 matrix, use the formula \(a(ei − fh) − b(di − fg) + c(dh − eg)\) where the elements of the matrix are arranged as \(\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\).
4. For matrices larger than 3x3, consider breaking them down into smaller matrices using the method of minors and cofactors, and then calculate the determinant recursively.
5. Verify the calculations at each step to ensure accuracy, as errors in earlier steps can propagate and lead to incorrect results.

Key Concepts

Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible (non-zero determinant) or singular (zero determinant). Determinants can be calculated using various methods, including expansion by minors or row reduction.

Matrix Operations

Matrix operations, including addition, subtraction, and multiplication, are fundamental in linear algebra. Understanding how to manipulate matrices is essential for evaluating determinants, as the properties of matrices directly influence the calculation of their determinants. For example, the determinant of a product of matrices equals the product of their determinants.

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a matrix by breaking it down into smaller matrices. This technique involves selecting a row or column, multiplying each element by its corresponding cofactor (which is the determinant of the submatrix formed by removing the row and column of that element), and summing these products. It is particularly useful for larger matrices.