Problem 7 - Precalculus with Limits

Question

In Exercises 5-20, find the determinant of the matrix. \(\left[ \begin{array}{r} 8 & 4 \\ 2 & 3 \end{array} \right]\)

Step-by-Step Solution

Answer

The determinant of the matrix is 16

Step 1: Identify the matrix elements

The matrix elements are identified as \(a = 8\), \(b = 4\), \(c = 2\), and \(d = 3\)

Step 2: Apply the formula for determinant of 2x2 matrix

Apply the formula for determinant which is \(ad - bc\). Substitute the values of a,b,c,d into the formula: \(8*3 - 2*4\)

Step 3: Calculate the determinant

Perform the multiplication and subtraction: \(24 - 8 = 16\)

Key Concepts

Determinant of a 2x2 Matrix

Understanding the determinant of a 2x2 matrix is integral in linear algebra as it provides information about the matrix's properties. To find the determinant of a matrix such as
\[\begin{equation}\left[ \begin{array}{rr} 8 & 4 \ 2 & 3 \end{array} \right]\end{equation}\]
, we use a simple formula. If we denote this matrix as \[\begin{equation}\left[ \begin{array}{cc}a & b \c & d \end{array} \right]\end{equation}\]
, its determinant is calculated as \[\begin{equation}ad - bc\end{equation}\]
. By plugging our identified values from the exercise (a=8, b=4, c=2, d=3), the determinant is \[\begin{equation}8*3 - 2*4 = 24 - 8 = 16\end{equation}\]
. A non-zero determinant, such as 16 in this case, indicates that the matrix is invertible and its columns are linearly independent.

Matrix Elements

Matrix elements are the individual numbers that fill in the grid of a matrix and are denoted by their position within that grid. In this exercise, the matrix elements are denominated as a, b, c, and d, where 'a' is at the top left, 'b' at the top right, 'c' at the bottom left, and 'd' at the bottom right position in the 2x2 matrix. Understanding the positioning is crucial for not only finding the determinant but also for performing more advanced matrix operations.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, spaces, and linear mappings between them. It's foundational for understanding systems of linear equations, transformations, and more complex structures like vector spaces and matrices. In the context of our exercise, linear algebra principles come into play when we discuss concepts like the determinant, which has applications in finding the inverse of a matrix and determining the solvability of a system of equations.

Matrix Operations

Matrix operations include a variety of computations that can be performed on matrices. Basic operations include addition, subtraction, scalar multiplication, and matrix multiplication. However, more complex operations also exist like finding the determinant, the transpose of a matrix, and the inverse, if it exists. In our case, the calculation of the determinant using the formula for a 2x2 matrix is an operation that requires both multiplication and subtraction of the matrix's elements in a specific order.