Problem 7 - College Algebra and Calculus: An Applied Approach

Question

Find the determinant of the matrix. $$ \left[\begin{array}{rr} 9 & 3 \\ 12 & 4 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the given matrix is 0.

Step 1: Understand the matrix

The given matrix is \[ \left[\begin{array}{rr} 9 & 3 \ 12 & 4 \end{array}\right] \]. In this matrix, the elements are a = 9, b = 3, c = 12, d = 4.

Step 2: Apply the formula

The determinant of a 2x2 matrix is given by the formula \( \text{det}(A) = a*d - b*c \). Apply this formula to the given matrix, substituting the letters with their respective elements: 9*4 - 3*12.

Step 3: Simplify the expression

Simplify the expression 9*4 - 3*12 to get the determinant. Calculating gives: 36 - 36.

Step 4: Final Determinant

Further simplifying the expression 36 - 36 gives a determinant of 0.

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple structure in mathematical terms, characterized by two rows and two columns, giving us a total of four elements. These matrices form a part of many mathematical and practical applications. When you see a 2x2 matrix written like: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \],this arrangement represents each number's position within the matrix. Here, \(a\), \(b\), \(c\), and \(d\) are individual numbers or expressions placed in specific slots. This structure is more than just a collection of numbers. It's a foundation for more complex calculations. Its simplicity makes it a perfect starting point for understanding larger matrices. You will often encounter 2x2 matrices when starting out with linear algebra, as they help introduce you to core concepts while maintaining manageability.

Linear Algebra

Linear algebra is a branch of mathematics dedicated to the study of vectors, matrices, and linear transformations. At its core, linear algebra is about understanding the way we can manipulate linear equations and functions.
The basics you should know include:

• Vectors: These are objects that have both a magnitude and a direction. They can be thought of as points, arrays of numbers, or arrows in a coordinate space.
• Matrices: These are arrays of numbers laid out in rows and columns, which can represent multiple linear equations. They are essential in solving systems of linear equations efficiently.
• Linear Transformations: These are functions that map vectors to other vectors, preserving vector addition and scalar multiplication.

Linear algebra plays a critical role in various fields such as computer science, physics, and engineering. It is the backbone of many algorithms and software applications used in today's technology.

Determinant Formula

The determinant of a matrix provides a lot of information about the matrix. For a 2x2 matrix, the determinant can tell us about the matrix's invertibility and the volume scaling factor when using the matrix as a transformation.For a 2x2 matrix: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \],the determinant is calculated using the formula:\[ \text{det}(A) = ad - bc \].
Here's how it works:

• Multiply the top left element \( a \) by the bottom right element \( d \).
• Subtract the product of the top right element \( b \) and the bottom left element \( c \).

In our specific case, using the numbers from the matrix \[ \begin{bmatrix} 9 & 3 \ 12 & 4 \end{bmatrix} \], we have:

• \( a \times d = 9 \times 4 = 36 \)
• \( b \times c = 3 \times 12 = 36 \)
• \( ext{det}(A) = 36 - 36 = 0 \)

A determinant of 0 indicates that the matrix is singular, which means it does not have an inverse. This is crucial when solving systems of linear equations, as it implies that the system might be either inconsistent or dependent.