Problem 16 - Intermediate Algebra

Question

Evaluate each of the following determinants. $$ \left|\begin{array}{rr} -9 & -7 \\ -6 & -4 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the given matrix is -6.

Step 1: Understand the Determinant of a 2x2 Matrix

The formula to find the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( ad - bc \). This equation essentially finds the product of the diagonal elements and subtracts the product of the off-diagonal elements.

Step 2: Identify the Elements

In the given matrix \(\begin{array}{rr} -9 & -7 \ -6 & -4 \end{array}\), the elements are identified as follows: \( a = -9 \), \( b = -7 \), \( c = -6 \), and \( d = -4 \).

Step 3: Substitute Values into the Determinant Formula

Substitute the identified values into the determinant formula: \( ad - bc = (-9)(-4) - (-7)(-6) \).

Step 4: Calculate the Product of Diagonals

Calculate \((-9)(-4)\), which is equal to \(36\).

Step 5: Calculate the Product of Off-Diagonals

Calculate \((-7)(-6)\), which is equal to \(42\).

Step 6: Compute the Determinant

Apply the operations identified: \(36 - 42 = -6\). Thus, the determinant of the matrix is \(-6\).

Key Concepts

Matrix Algebra

Matrix algebra is a fundamental component of mathematics that deals with the study of matrices and the operations that can be performed on them. It is a crucial topic not just for students of mathematics, but also for anyone working in fields that utilize large and complex data sets. In essence, a matrix is a rectangular array of numbers arranged in rows and columns. The basic operations in matrix algebra involve addition, subtraction, and most importantly for our topic, multiplication.

Multiplying matrices involves combining the elements of matrices to produce a single number or another matrix. For 2x2 matrices, this multiplication is particularly important when calculating the determinant. The determinant helps in numerous applications such as solving systems of linear equations, and understanding properties of linear transformations. It is calculated specifically for square matrices and reflects various properties of the matrix, such as invertibility.

Using matrix algebra efficiently requires understanding how different operations influence the values within the matrix and their implications on larger mathematical models and systems.

Linear Algebra

Linear algebra, the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces, is foundational for understanding complex systems. It provides a structured framework for addressing problems in various disciplines like computer science, engineering, physics, and more.

In the context of linear algebra, matrices are utilized to represent linear transformations. A linear transformation is essentially any function that preserves vector addition and scalar multiplication. Matrices simplify understanding these transformations by providing a compact form to work with. By analyzing the determinant of a matrix, we gain insight into certain characteristics of the transformation, like whether it is invertible or its impact on volume scaling in geometric transformations.

Linear algebra is essentially about understanding how to work with these numerical structures (like matrices) to perform operations that can solve real-world problems in a variety of fields.

Determinants Calculation

One of the key processes in linear and matrix algebra is the calculation of determinants, specifically for 2x2 matrices. Understanding how to compute a determinant is crucial because it helps assess whether a matrix is invertible and is also used in various other applications like calculating eigenvalues and eigenvectors.

To calculate the determinant of a 2x2 matrix, we use a straightforward formula: if the matrix is \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \], the determinant is computed as \( ad - bc \). This formula involves two crucial steps:

• Multiply the elements of the main diagonal (\( a \) and \( d \)).
• Subtract the product of the off-diagonal elements (\( b \) and \( c \)).

In practical terms, determining the determinant can confirm whether a system of equations has unique solutions. If the determinant is non-zero, the matrix is invertible, meaning the system has a unique solution. In contrast, a determinant of zero implies no unique solutions, often indicating either none or infinitely many solutions. This highlights the importance of determinants in both matrices and broader mathematical applications.