Evaluate each determinant in Exercises 1–10.
$\(\begin{vmatrix}\)5 & 7 \\2 & 3\(\end{vmatrix}\)$

Verified step by step guidance

1. Identify the matrix given as a 2x2 matrix: \(\begin{bmatrix} 5 & 7 \\ 2 & 3 \end{bmatrix}\).
2. Recall the formula for the determinant of a 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\), which is \(ad - bc\).
3. Assign the values from the matrix to the variables: \(a = 5\), \(b = 7\), \(c = 2\), and \(d = 3\).
4. Substitute these values into the determinant formula: \(5 \times 3 - 7 \times 2\).
5. Simplify the expression to find the determinant value (do not calculate the final number here).

Key Concepts

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. It is a scalar value that provides important information about the matrix, such as whether it is invertible. For the given matrix [[5, 7], [2, 3]], the determinant is (5)(3) - (7)(2).

Matrix Notation and Elements

A matrix is a rectangular array of numbers arranged in rows and columns. Each element is identified by its row and column position. Understanding how to read and identify elements in a matrix is essential for performing operations like finding determinants.

Properties and Applications of Determinants

Determinants help determine if a matrix is invertible (non-zero determinant) and are used in solving systems of linear equations, finding area or volume transformations, and understanding linear transformations. Recognizing these applications helps contextualize why calculating determinants is important.