Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}8 & -2 & -4 \\\7 & 0 & 3 \\\5 & -1 & 2\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 54.

Step 1: Identify the Matrix and Calculate the Determinant

We begin with the matrix:\[\begin{bmatrix}8 & -2 & -4 \7 & 0 & 3 \5 & -1 & 2\end{bmatrix}\] We want to find its determinant using the formula for a 3x3 matrix:\[\operatorname{det}\begin{bmatrix}a & b & c \d & e & f \g & h & i\end{bmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)\]For our matrix, \(a = 8, b = -2, c = -4, d = 7, e = 0, f = 3, g = 5, h = -1, i = 2\).

Step 2: Compute the First Term Contribution

Calculate the contribution of the first term, \(a(ei - fh)\):\[\begin{aligned}ei - fh &= 0 \cdot 2 - 3 \cdot (-1) \&= 0 + 3 = 3\end{aligned}\]So, the first term is:\[8 \times 3 = 24\]

Step 3: Compute the Second Term Contribution

Calculate the contribution of the second term, \(-b(di - fg)\):\[\begin{aligned}di - fg &= 7 \cdot 2 - 3 \cdot 5 \&= 14 - 15 = -1\end{aligned}\]So, the second term is:\[-(-2) \times (-1) = 2\]

Step 4: Compute the Third Term Contribution

Calculate the contribution of the third term, \(c(dh - eg)\):\[\begin{aligned}dh - eg &= 7 \cdot (-1) - 0 \cdot 5 \&= -7 - 0 = -7\end{aligned}\]So, the third term is:\[-4 \times (-7) = 28\]

Step 5: Combine All Contributions

Now, add up all the contributions from the three terms:\[24 + 2 + 28 = 54\]

Step 6: Final Result: Determinant Value

The determinant of the given matrix is 54.

Key Concepts

3x3 Matrix

A 3x3 matrix is simply a square grid of numbers with three rows and three columns. It is a fundamental structure in both mathematics and computer science. This type of matrix can represent different systems and transformations in multidimensional space.
Consider the matrix given in our exercise, which looks like this:\[ \begin{bmatrix} 8 & -2 & -4 \ 7 & 0 & 3 \ 5 & -1 & 2 \end{bmatrix} \]Every position in a matrix is critical as it represents data in our system of linear equations. Each element can be referred to using a pair of indices, like \(a_{ij}\), where \(i\) is the row number and \(j\) is the column number.
In applications such as computer graphics and physics, a 3x3 matrix helps in transformations like rotation and scaling operations. Understanding the structure is essential for performing operations like addition, subtraction, and multiplication, both by other matrices and by scalar values. In this context, calculating the determinant of a 3x3 matrix plays a crucial role, as it aids in understanding the matrix’s properties.

Linear Algebra

Linear algebra is an area of mathematics dedicated to the study of vectors, vector spaces, and linear transformations. It is foundational for various fields, from quantum mechanics to economics, due to its widespread applications.
Matrices, like our 3x3 matrix, are central to linear algebra because they allow us to work with linear equations in an organized way. For instance:

• They help solve systems of linear equations by representing the coefficients of variables.
• Matrices can represent and solve large systems efficiently, especially with computational tools.
• Matrix operations are used to determine whether a system has a unique solution, infinite solutions, or no solution.

Another fundamental concept in linear algebra is the determinant, a value that provides essential insights into matrix properties, such as invertibility and the solutions to a matrix equation. By engaging with these concepts, students can develop a deeper understanding of how systems are structured and solved in linear algebra.

Matrix Determinant Formula

The determinant of a matrix is a special number that you compute from its elements. For a 3x3 matrix, the determinant can reveal much about the matrix's behavior and properties. The formula to find the determinant of a 3x3 matrix is:\[\operatorname{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]where the matrix A is given by:\[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]
Calculating the determinant involves breaking the 3x3 matrix down into smaller elements and calculating the effects of each element through this predefined method.

• The term \(a(ei - fh)\) is calculated by focusing on the row and column excluding 'a' and multiplying the respective minors.
• The second term \(-b(di - fg)\) considers the matrix elements in a similar manner and includes a negative sign due to the pattern of cofactors.
• The final term \(c(dh - eg)\) completes the expansion of minors needed for full computation.

Together, these calculations provide insights into the matrix's geometric interpretation, like the volume scaling factor it can determine in space transformations. Understanding this formula allows students to analyze more complex systems and appreciate the elegance of matrix mathematics.