Evaluate each determinant. See Example 1.

Verified step by step guidance

1. Identify the matrix from which the determinant needs to be calculated. If the matrix is given in the problem, write it down in square matrix form.
2. Check the order of the matrix. If it is a 2x2 matrix, use the formula \(ad - bc\) where \(a, b, c,\) and \(d\) are the elements of the matrix. If it is a 3x3 matrix or larger, consider using cofactor expansion or other determinant-solving methods.
3. For a 2x2 matrix, substitute the values of the elements into the determinant formula. For a 3x3 matrix, break it down into smaller 2x2 matrices and calculate the determinant for each, then sum them up according to the rules of cofactor expansion.
4. Simplify the expressions obtained in the previous step to get the value of the determinant.
5. Interpret the result. A zero determinant indicates that the matrix is singular, meaning it does not have an inverse. A non-zero determinant means the matrix is invertible.

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 row reduction, cofactor expansion, or leveraging properties of determinants.

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 these operations can simplify calculations. For instance, the determinant of a product of matrices equals the product of their determinants, which can be useful in complex evaluations.

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 where direct computation is cumbersome.