Evaluate each determinant in Exercises 1–10. 5 72 3

Verified step by step guidance

1. Identify the matrix as a 2x2 matrix with elements a=5, b=7, c=2, d=3.
2. Recall the formula for the determinant of a 2x2 matrix: det(A) = ad - bc.
3. Substitute the values into the formula: det(A) = (5)(3) - (7)(2).
4. Calculate the products: (5)(3) and (7)(2).
5. Subtract the second product from the first to find the determinant.

Key Concepts

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix is calculated using the formula |A| = ad - bc, where A = |a b| and |c d|. For the matrix given, the determinant is computed as (5)(3) - (7)(2), which simplifies to 15 - 14 = 1. The determinant provides important information about the matrix, including whether it is invertible.

Matrix Representation

A matrix is a rectangular array of numbers arranged in rows and columns. In this case, the matrix is represented as two rows and two columns, with the first row containing the numbers 5 and 7, and the second row containing 2 and 3. Understanding how to read and interpret matrices is essential for performing operations such as finding determinants.

Properties of Determinants

Determinants have several key properties, including that a matrix is invertible if and only if its determinant is non-zero. Additionally, the determinant changes sign if two rows are swapped, and if a row is multiplied by a scalar, the determinant is also multiplied by that scalar. These properties are crucial for understanding the implications of the determinant in linear algebra.