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 a 2x2 matrix, it will look like this: \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\).
2. Apply the formula for the determinant of a 2x2 matrix, which is \(ad - bc\). Here, \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix.
3. Substitute the actual values of \(a\), \(b\), \(c\), and \(d\) from your matrix into the formula.
4. Perform the multiplication for both \(ad\) and \(bc\).
5. Subtract the product \(bc\) from the product \(ad\) 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 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.