Problem 33 - Precalculus

Question

Evaluate each \(3 \times 3\) determinant. $$\left|\begin{array}{rrr} 2 & 1 & -5 \\ 3 & 0 & -1 \\ 4 & 0 & 7 \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant of the matrix is \(-25\).

Step 1: Identify the Formula for a 3x3 Determinant

For a 3x3 matrix \( \left[ \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \right] \), the determinant \( \Delta \) is calculated as follows:\[ \Delta = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Step 2: Substitute Matrix Elements into the Formula

Given the matrix \( \left[ \begin{array}{ccc} 2 & 1 & -5 \ 3 & 0 & -1 \ 4 & 0 & 7 \end{array} \right] \), identify each element with the formula:\( a = 2, b = 1, c = -5, d = 3, e = 0, f = -1, g = 4, h = 0, i = 7 \). Now substitute these values into the determinant formula.

Step 3: Calculate Each Part of the Formula

Calculate each sub-expression within the determinant formula:1. \( ei - fh = (0 \times 7) - (-1 \times 0) = 0 2. \) 2. \( di - fg = (3 \times 7) - (-1 \times 4) = 21 + 4 = 25 2. \) 3. \( dh - eg = (3 \times 0) - (0 \times 4) = 0 2. \) Substitute these results: \( \Delta = 2(0) - 1(25) - 5(0) \).

Step 4: Solve the Final Expression

Substitute the simplified values into the expression from the previous step:\[ \Delta = 2 \times 0 - 1 \times 25 - 5 \times 0 = 0 - 25 - 0 = -25 \]

Step 5: Interpret the Result

The determinant of the given matrix is \(-25\). This value indicates how the matrix transformation affects the area or volume in this mathematical space.

Key Concepts

Matrix Transformations

Matrix transformations are operations that take a set of points and transform them into another set of points. These operations are often represented using matrices. For a clearer picture, imagine a transformation as instructions on translating or rotating a shape in a two-dimensional or three-dimensional plane.

• Translation: Moving a shape to a different location without altering its orientation or size.
• Rotation: Spinning the shape around a specific point.
• Scaling: Making the shape larger or smaller.

When a transformation is applied using a matrix, each point in the space is multiplied by the matrix. The determinant of this matrix tells us something important: - If the determinant is zero, the transformation squashes the space into a lower dimension. - If the determinant is negative, the transformation includes a reflection. - The absolute value of the determinant describes the scale or change in volume or area the transformation causes.

3x3 Matrix

A 3x3 matrix is a collection of nine numbers arranged into three rows and three columns. Each number in a matrix is called an element, and these elements are used in various mathematical computations, including solving systems of equations and transforming geometric figures. Understanding the layout: - The first index indicates the row, and the second index indicates the column. For example, in a 3x3 matrix, the element in the second row and first column is often denoted as \(d\).Working with a 3x3 matrix involves basic operations such as addition, subtraction, and multiplication. However, one of the most critical operations is finding the determinant, which points to the matrix's invertibility and the kind of transformation it represents. The determinant also plays a crucial role in calculating the inverse of the matrix, which is required for solving linear equations.

Linear Algebra

Linear algebra is a branch of mathematics focused on vector spaces and linear transformations. It's essential in various scientific fields, including physics, computer science, and engineering. Key concepts in linear algebra include:

• Vectors: Objects that have direction and magnitude, represented as points or arrows in space.
• Matrices: Arrays of numbers that can represent linear equations or transformations.
• Determinants: Scalars that provide information about a matrix's properties, such as invertibility and volume transformation.

Linear algebra enables you to solve systems of linear equations efficiently using matrices. This is pivotal in fields as varied as 3D modeling, computer graphics, and machine learning. Understanding the determinants and transformations covered in this area helps in discerning how changes in one vector space may impact another.

Mathematical Space

Mathematical space, or just space in mathematics, refers to a set of elements that have geometric or physical interpretation, like points or vectors. There are different types of mathematical spaces, each with its unique properties.
One common space encountered is Euclidean space, where ordinary geometry is applied to shapes and figures we see around us. When matrices and transformations come into play, these shapes can be expertly manipulated:

• Two-dimensional space: A flat plane where transformations govern the positions of shapes.
• Three-dimensional space: Here, you see depth and volume, adding complexity to transformations in areas such as modeling and physics.

Engaging with mathematical spaces is foundational when exploring linear algebra topics. The specific role of determinants within these spaces is to denote how transformations affect areas and volumes, offering profound insights into real and theoretical phenomena.