Problem 28 - Algebra 2

Question

Evaluate each determinant. $$ \left|\begin{array}{cc}{-\frac{1}{2}} & {2} \\ {-2} & {8}\end{array}\right| $$

Step-by-Step Solution

Answer

\(-\frac{1}{2} \cdot 8 - 2 \cdot -2 = -4 + 4 = 0\)

Step 1: Understand the Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix \(A\) with elements \(a, b, c, d\) arranged as:\[\begin{pmatrix}a & b\ c & d\end{pmatrix}\] is calculated as \(ad - bc\).

Step 2: Identify Matrix Elements

Identify the elements of the given matrix. In this case:\[a = -\frac{1}{2}, b = 2, c = -2, d = 8\]

Step 3: Apply the Determinant Formula

Apply the formula to the matrix elements:\[\text{Determinant} = (-\frac{1}{2} \cdot 8) - (2 \cdot -2)\]

Key Concepts

Matrix Elements

When working with matrices, it's crucial to understand the term matrix elements. Matrix elements are the individual numbers or expressions contained within the matrix. Each element has a specific position, identified by its row and column numbers. To imagine it more vividly, think of the matrix as a spreadsheet where each cell is a matrix element.

In a 2x2 matrix, we name the elements as follows: For a matrix \( A \) with elements \( a, b, c, d \) arranged as:\[\begin{pmatrix}a & b\c & d\end{pmatrix}\], \( a \) is the element in the first row and first column, \( b \) is in the first row and second column, \( c \) in the second row and first column, and \( d \) in the second row and second column. It's essential to identify these elements correctly to perform any matrix-related operations, including finding the determinant.

Determinant Calculation

The process of determinant calculation is a fundamental concept in linear algebra. For a 2x2 matrix, the determinant provides us with a single number that can tell us a lot about the matrix, such as whether it's invertible or what its scaling factor is. The determinant is calculated using a simple formula: For matrix \( A \) with elements \( a, b, c, d \) as described earlier, the determinant \( \text{Det}(A) \) is \( ad - bc \).

Applying this formula to the provided exercise, we substitute the matrix elements into the formula to get \( \text{Determinant} = (-\frac{1}{2} \cdot 8) - (2 \cdot -2) \), thus calculating the determinant by multiplying the elements and subtracting the product of the diagonals. This crucial step can inform us whether the matrix can be used to solve systems of equations, how it transforms space, and more.

Linear Algebra

The field of linear algebra is vast and critical to various areas of mathematics and science, including computer science, physics, and engineering. It deals with vectors, vector spaces, linear mappings, and systems of linear equations. Determinants are a small, yet mighty part of linear algebra, and they play a vital role in understanding whether a system of linear equations has a unique solution, no solution, or infinitely many solutions.

In linear algebra, the concepts of matrix transformations are often visualized as functions that stretch, squeeze, rotate, or reflect shapes in a given space. The determinant helps to understand these transformations by describing how the area or volume is affected. For example, if the determinant is zero, the transformation squashes the space into a lower dimension, which is closely related to the concept of linear independence and the span of vectors.