Evaluate each determinant in Exercises 1–10.
$\(\begin{vmatrix}\)-4 & 1 \\5 & 6\(\end{vmatrix}\)$

Verified step by step guidance

1. Identify the matrix given as \( \begin{pmatrix} -4 & 1 \\ 5 & 6 \end{pmatrix} \).
2. Recall the formula for the determinant of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by \( \text{det} = ad - bc \).
3. Substitute the values from the matrix into the formula: \( a = -4, b = 1, c = 5, d = 6 \).
4. Calculate the product of the diagonal elements: \( a \times d = (-4) \times 6 \).
5. Calculate the product of the off-diagonal elements: \( b \times c = 1 \times 5 \), then subtract this from the previous product to find the determinant.

Key Concepts

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This scalar value helps determine properties like invertibility and area scaling in linear transformations.

Matrix Notation and Elements

Understanding matrix notation involves recognizing the position of elements: 'a' and 'b' are in the first row, 'c' and 'd' in the second. Correctly identifying these values is essential for accurate determinant calculation.

Applications of Determinants

Determinants are used to solve systems of linear equations, find matrix inverses, and analyze geometric transformations. Evaluating determinants is a foundational skill in linear algebra and college algebra.