Problem 23 - A Graphical Approach to College Algebra

Question

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}0.4 & -0.8 & 0.6 \\\0.3 & 0.9 & 0.7 \\\3.1 & 4.1 & -2.8\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is -5.5.

Step 1: Understand the Determinant Formula for 3x3 Matrices

The determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is given by the formula: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] We will apply this formula to find the determinant of the given matrix.

Step 2: Identify the Elements of the Matrix

Identify the elements of the matrix: \( a = 0.4, b = -0.8, c = 0.6 \) \( d = 0.3, e = 0.9, f = 0.7 \) \( g = 3.1, h = 4.1, i = -2.8 \)

Step 3: Calculate Minor Determinants

Compute the minor determinants as follows: \( ei - fh = (0.9)(-2.8) - (0.7)(4.1) = -2.52 - 2.87 = -5.39 \) \( di - fg = (0.3)(-2.8) - (0.7)(3.1) = -0.84 - 2.17 = -3.01 \) \( dh - eg = (0.3)(4.1) - (0.9)(3.1) = 1.23 - 2.79 = -1.56 \)

Step 4: Substitute and Compute the Determinant

Substitute the values into the determinant formula: \[ \text{det}(A) = 0.4(-5.39) - (-0.8)(-3.01) + 0.6(-1.56) \] Calculate each term: - \( 0.4 \times -5.39 = -2.156 \) - \( -0.8 \times -3.01 = -0.8 \times (-3.01) = 2.408 \) (note the negative sign before \(b\) cancels part of the expression) - \( 0.6 \times -1.56 = -0.936 \). Add these results to find the determinant: \[ -2.156 - 2.408 - 0.936 = -5.5 \]

Step 5: Conclusion

The determinant of the matrix is \(-5.5\). It indicates the signed volume of the parallelepiped formed by vectors corresponding to the matrix rows.

Key Concepts

3x3 Matrix

A 3x3 matrix is a grid of numbers arranged in three rows and three columns. This arrangement is a central concept in linear algebra and is written in a matrix form. A typical 3x3 matrix looks like this: \[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\] Here, each letter represents an element or number in the matrix. In the original problem, the elements are specific numbers: - First row: 0.4, -0.8, 0.6 - Second row: 0.3, 0.9, 0.7 - Third row: 3.1, 4.1, -2.8These elements are part of a 3x3 matrix used to compute something important called a "determinant." The determinant provides valuable information about the matrix, such as scaling factors, orientation of vectors, and more. Understanding a 3x3 matrix is crucial before diving into calculations like finding the determinant.

Matrix Algebra

Matrix algebra involves various operations, such as addition, subtraction, and multiplication, which are unique compared to regular arithmetic. These operations follow specific rules, making matrix algebra a powerful tool in solving complex problems. It often symbolizes systems of equations, transformations in space, and more. One important operation is the multiplication between matrices or a matrix and a scalar (a single number). This operation must follow specific rules that define how each element of the result is calculated. Practicing these operations builds strong intuition for understanding matrices. The determinant calculation is a specialized operation within matrix algebra. It helps in finding out whether a matrix is invertible, meaning it has an inverse, or determining characteristics about geometric transformations represented by the matrix. Matrix algebra is essential not just for determinants but for broader applications in computer science, physics, and engineering.

Determinant Formula

The determinant is a scalar value derived from a square matrix, such as a 3x3 matrix. The determinant formula for a 3x3 matrix involves arithmetic operations on its elements to determine a single numeric value. The formula for calculating the determinant of a matrix \(A\) with elements \(a, b, c, d, e, f, g, h, i\) is:\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]This equation might look complex at first glance, but breaking it down step-by-step simplifies understanding:

• Calculate three minor determinants: \(ei - fh\), \(di - fg\), and \(dh - eg\).
• Substitute these minor determinants into the main formula.
• Simplify the expressions to find the determinant value.

In the exercise, the calculated determinant is - \(-5.5\).The value of the determinant can indicate several things, such as the orientation and volume of a geometric space defined by the matrix. In systems of equations, a non-zero determinant often means there's a unique solution, further reflecting its critical role in various mathematical fields.