Problem 13 - Algebra 2

Question

Find the value of each determinant. $$ \left|\begin{array}{rr}{-6} & {-2} \\ {8} & {5}\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is -14.

Step 1: Understanding the 2x2 Determinant Formula

For any 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated using the formula: \( ad - bc \). We identify \( a, b, c, \) and \( d \) from the given matrix.

Step 2: Identifying the Elements of the Matrix

Given the matrix \( \begin{pmatrix} -6 & -2 \ 8 & 5 \end{pmatrix} \), we identify the elements as: \( a = -6 \), \( b = -2 \), \( c = 8 \), and \( d = 5 \).

Step 3: Calculate the Product ad

Multiply \( a \) and \( d \) to get \( ad = (-6) \times 5 = -30 \).

Step 4: Calculate the Product bc

Multiply \( b \) and \( c \) to get \( bc = (-2) \times 8 = -16 \).

Step 5: Apply the Determinant Formula

Substitute the products we calculated into the determinant formula: \( ad - bc = -30 - (-16) \).

Step 6: Solve the Deteminant Expression

Simplify the expression \( -30 + 16 \) to obtain \( -14 \). This is the value of the determinant.

Key Concepts

2x2 matrix

A 2x2 matrix is a simple and commonly used type of matrix in linear algebra. It consists of two rows and two columns. Each position in this grid contains a number, known as an element of the matrix. An example is given by the matrix \( \begin{pmatrix} -6 & -2 \ 8 & 5 \end{pmatrix} \). Here, the elements are identified as follows: the first row contains \(-6\) and \(-2\), and the second row contains \(8\) and \(5\).
Understanding a 2x2 matrix is foundational for grasping more complex matrix structures and operations. These matrices are often used to represent and solve linear equations, which are equations of straight lines. Translating real-world problems into 2x2 matrices can often simplify problem-solving. In any matrix, identifying the positions of elements is key, such as "row one, column two" correlating to the location of \(-2\).
It's essential when working with matrices to keep track of these elements as they define the operations we perform, such as finding the determinant.

determinant formula

The determinant of a 2x2 matrix is a specific number calculated from its elements. For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is given by the formula \( ad - bc \). This formula is crucial because it provides insights into certain properties of the matrix.

• If the determinant is zero, the matrix is called "singular," meaning it has no inverse.
• A non-zero determinant indicates the matrix is "non-singular" and an inverse exists.

The determinant has geometric interpretations as well, such as scaling factor or area when applying the matrix to vector spaces. For our example matrix \( \begin{pmatrix} -6 & -2 \ 8 & 5 \end{pmatrix} \), first, we identify the elements: \( a = -6 \), \( b = -2 \), \( c = 8 \), \( d = 5 \). Then, calculate \( ad = (-6) \times 5 = -30 \) and \( bc = (-2) \times 8 = -16 \). Substituting in the formula gives \( -30 - (-16) = -14 \). This shows how 2x2 matrices allow quick determination of important characteristics through this simple computation.

linear algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations—operations that can be represented using matrices. Within this field, matrices like our 2x2 matrix \( \begin{pmatrix} -6 & -2 \ 8 & 5 \end{pmatrix} \) are tools used for solving systems of linear equations, mapping geometric transformations like rotations and reflections, and more.
Matrices and determinants play central roles in linear algebra for tasks that require computing the area of parallelograms, determining system solvability, and more. The determinant of a matrix like \( \begin{pmatrix} -6 & -2 \ 8 & 5 \end{pmatrix} \) is particularly important for understanding the matrix's properties—how it scales and transforms space.
Using linear algebra, we can model and solve real-world problems in physics, engineering, and computer science, making mastery of concepts like matrix determinant foundational to advancing in these fields.

matrix operations

Performing operations on matrices is a key aspect of using them effectively in solving mathematical problems. With our given 2x2 matrix \( \begin{pmatrix} -6 & -2 \ 8 & 5 \end{pmatrix} \), we can execute several operations: addition, multiplication, and finding determinants.

• Matrix addition involves adding corresponding elements from two matrices of the same size.
• Matrix multiplication, a bit more complex, involves row-by-column multiplication between two matrices.
• The determinant computation, as shown, condenses the behavior of a matrix into a single number, influencing the feasibility of other operations, like finding inverses.

In practical applications, such as physics or computer graphics, these operations are crucial. For example, in graphics, matrices help rotate objects or transform images consistently. Understanding these basic matrix operations allows one to start working on more complex mathematical models that simulate real-world scenarios.