In Exercises 46–51, evaluate each determinant.

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 along that row or column. The formula is \( \text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} \) for a 3x3 matrix, where \( C_{ij} \) is the cofactor of element \( a_{ij} \).
4. Calculate the cofactors for each element in the chosen row or column. The cofactor \( C_{ij} \) is given by \( (-1)^{i+j} \times M_{ij} \), where \( M_{ij} \) is the minor of \( a_{ij} \), found by removing the i-th row and j-th column from the matrix.
5. Sum the products of each element in the chosen row or column with its corresponding cofactor 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 and the volume scaling factor of the linear transformation represented by the matrix. The determinant can be calculated using various methods, including cofactor expansion and row reduction.

Square Matrices

A square matrix is a matrix with the same number of rows and columns, denoted as n x n. Determinants are only defined for square matrices, as they represent linear transformations in n-dimensional space. Understanding the properties of square matrices, such as rank and eigenvalues, is essential for evaluating 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.