Evaluate each determinant in Exercises 1–10. - 4 15 6

Verified step by step guidance

1. Identify the elements of the 2x2 matrix: a = -4, b = 1, c = 5, d = 6.
2. Recall the formula for the determinant of a 2x2 matrix: det(A) = ad - bc.
3. Substitute the identified elements into the formula: det(A) = (-4)(6) - (1)(5).
4. Calculate the product of a and d: (-4)(6).
5. Calculate the product of b and c: (1)(5).

Key Concepts

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix is calculated using the formula det(A) = ad - bc, where A = |a b| and |c d|. For the matrix given, the values are a = -4, b = 1, c = 5, and d = 6. This determinant provides important information about the matrix, such as whether it is invertible and the area of the parallelogram formed by its column vectors.

Properties of Determinants

Determinants have several key properties, including that the determinant of a matrix is zero if the rows or columns are linearly dependent. Additionally, swapping two rows or columns changes the sign of the determinant, while multiplying a row or column by a scalar multiplies the determinant by that scalar. Understanding these properties is essential for simplifying calculations and analyzing matrices.

Applications of Determinants

Determinants are used in various applications, including solving systems of linear equations, finding the area of geometric shapes, and determining the invertibility of matrices. In the context of linear algebra, they help in understanding the behavior of linear transformations and can be used in calculus for evaluating integrals in multiple dimensions.