In Exercises 46–51, evaluate each determinant.

Verified step by step guidance

1. Step 1: Identify the size of the determinant. If it's a 2x2 matrix, use the formula \( ad - bc \) for a matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).
2. Step 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. Step 3: For cofactor expansion, choose a row or column (usually the one with the most zeros for simplicity) and expand the determinant along that row or column. Multiply each element by its cofactor and sum the results.
4. Step 4: Calculate the cofactors by taking the determinant of the 2x2 matrix that remains after removing the row and column of the element being considered, and apply the sign based on the position \((-1)^{i+j}\).
5. Step 5: Sum the products of the elements and their corresponding cofactors to find the determinant of the 3x3 matrix.

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, and summing the results. It is particularly useful for larger matrices, allowing for a systematic approach to finding determinants.