Evaluate each determinant. See Example 3.

Verified step by step guidance

1. Identify the size of the determinant. If it's a 2x2 matrix, use the formula: \( \text{det}(A) = ad - bc \) for a matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).
2. If it's a 3x3 matrix, use the rule of Sarrus or the cofactor expansion method. For the rule of Sarrus, extend the first two columns of the matrix to the right and calculate the sum of the products of the diagonals from top left to bottom right, then subtract the sum of the products of the diagonals from bottom left to top right.
3. For cofactor expansion, choose a row or column (usually the one with the most zeros for simplicity) and expand the determinant using minors and cofactors. The formula is \( \text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} \) for the first row, where \( C_{ij} \) is the cofactor of element \( a_{ij} \).
4. Calculate the minors for each element in the chosen row or column. A minor is the determinant of the submatrix that remains after removing the row and column of the element.
5. Calculate the cofactors by applying the sign pattern \((-1)^{i+j}\) to each minor, and then sum the products of the original elements and their corresponding cofactors to find the determinant.

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 also be used to calculate the area or volume of geometric shapes defined by the matrix.

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 determinant of a matrix can be affected by these operations. Familiarity with these operations allows for the simplification of matrices before calculating 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, and summing the results. It is particularly useful for larger matrices, where direct computation of the determinant may be complex.