Problem 9 - College Algebra and Calculus: An Applied Approach

Question

Find the determinant of the matrix. $$ \left[\begin{array}{ll} 2 & 1 \\ 3 & 4 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the matrix is 5.

Step 1: Identify the elements of the matrix

The given matrix is \[ \begin{array}{ll} 2 & 1 \ 3 & 4 \end{array} \]. From this, we can identify the main diagonal elements as 2 and 4, and the secondary diagonal elements as 1 and 3.

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

The formula for the determinant of a 2x2 matrix is \[ad - bc\] where \(a\) and \(d\) are the elements of the main diagonal, and \(b\) and \(c\) are the elements of the secondary diagonal. Plugging in our values, we get \(2*4 - 1*3\)

Step 3: Compute the determinant

Finally, calculate the difference to get the determinant: \[2*4 - 1*3 = 8 - 3 = 5\]

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is a simple yet fundamental concept in linear algebra. It looks like a small table with two rows and two columns. Each number in these positions holds a specific meaning and can represent real-world data, like coordinates on a graph or coefficients in a system of equations. In matrix notation, it is presented as follows:
\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]
The letters \(a\), \(b\), \(c\), and \(d\) symbolize the numbers that fill each spot.

• The main diagonal consists of \(a\) and \(d\), going from the top-left to the bottom-right.
• The secondary diagonal includes \(b\) and \(c\), from top-right to bottom-left.

Understanding the layout of these elements helps us execute different mathematical operations like finding determinants.

Basics of Matrix Algebra

Matrix algebra is a branch of mathematics focusing on operations involving matrices. In this arena, matrices can be added, subtracted, multiplied, and more.
Here are some key operations:

• Addition/Subtraction: Combine matrices of the same size by adding or subtracting corresponding elements.
• Multiplication: Multiply matrices by following specific rules that involve the row-by-column rule, ensuring that the inner dimensions match.

These operations are similar to arithmetic with numbers but hold additional layers of complexity due to the spatial arrangement of data. In algebra, a matrix can represent entire systems of equations, allowing for solutions through elegant, systematic techniques. An essential operation in this context is finding the determinant, important for solving linear equations and more.

Exploring the Determinant Formula

The determinant of a 2x2 matrix is a special number that tells us a lot about the matrix's characteristics. It's a simple calculation yet vital in matrix algebra. The formula for the determinant \(D\) of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is:
\[D = ad - bc\]
Breaking it down:

• Multiply the elements of the main diagonal \(a\) and \(d\).
• Subtract the product of the secondary diagonal \(b\) and \(c\) from it.

This value can tell us if a matrix is invertible or not. A determinant of zero implies that the matrix cannot be inverted, which has implications in solving certain kinds of problems. The determinant also provides insights into the area scaling factor associated with the transformation that the matrix represents.