Problem 15 - Precalculus with Limits

Question

In Exercises 5-20, find the determinant of the matrix. \(\left[ \begin{array}{r} -3 & -2 \\ -6 & -1 \end{array} \right]\)

Step-by-Step Solution

Answer

The determinant of the given matrix is -9

Step 1: Identify Matrix Elements

Using the formula for the determinant of a 2x2 matrix, \(ad - bc\), we first identify the elements a, b, c, d in the matrix. These would be:\(a = -3, b = -2, c = -6, d = -1\).

Step 2: Substitute into the formula

Substitute the values of a, b, c, and d into the formula \((ad - bc)\):\(-3*-1 - (-2*-6)\)

Step 3: Final Computation

Simplify further:\(3 - 12 = -9\)

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix used in linear algebra. It contains two rows and two columns of numbers, making it a perfect starting point for understanding more complex matrix operations. In a 2x2 matrix, you will typically see it written in a block of four numbers, like this:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]Each of the numbers (or elements) in the matrix has a specific position, indicated by their row and column. For example, in the given matrix \[ \begin{bmatrix} -3 & -2 \ -6 & -1 \end{bmatrix} \], the top left number is in the first row and first column (\(a = -3\)).
You can think of each matrix element as a building block, contributing to its overall functionality. When working with matrices, understanding their structure and how they relate to each other is crucial for solving problems such as finding determinants or performing matrix multiplication.

What are Matrix Elements?

Matrix elements are the individual numbers inside the matrix that you work with when performing operations. Each element holds a unique position defined by its row and column number.

• For the matrix \( \begin{bmatrix} -3 & -2 \ -6 & -1 \end{bmatrix} \), the elements are \( a = -3 \), \( b = -2 \), \( c = -6 \), and \( d = -1 \).
• To find each element, identify its position: top row-first column (\(a\)), top row-second column (\(b\)), and so on.

These elements are essential when calculating the determinant of a matrix, which is a number that bundles up important information about the matrix.
Knowing how to identify and manipulate each element helps in numerous matrix operations in linear algebra, like solving systems of equations or transforming geometric shapes.

The Role of Linear Algebra

Linear algebra is a branch of mathematics focused on vectors, matrices, and the operations used to analyze linear equations and transformations. It plays a crucial role in multiple fields, such as physics, engineering, computer science, and finance.When dealing with matrices in linear algebra, you handle operations like addition, multiplication, and finding determinants, which are vital for solving real-world problems. For example, understanding the determinant of a matrix involves linear transformations and vector spaces which have broad applications.

• Determinants can help determine if a system of equations has a unique solution.
• They are also useful in computing the inverse of a matrix, if it exists.

In our example exercise, the determinant of the 2x2 matrix \[ \begin{bmatrix} -3 & -2 \ -6 & -1 \end{bmatrix} \] was found to be \(-9\). This value tells us important characteristics about the linear transformation that this matrix represents, such as if it preserves volume or changes orientation in a vector space.
Understanding linear algebraic principles allows students to build a strong foundation for future studies and practical applications.