Problem 54 - A Graphical Approach to College Algebra

Question

Use the determinant theorems to find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}4 & 8 & 0 \\\\-1 & -2 & 1 \\\2 & 4 & 3\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 0.

Step 1: Identify the Matrix

We are given a 3x3 matrix: \[\begin{bmatrix} 4 & 8 & 0 \ -1 & -2 & 1 \ 2 & 4 & 3 \end{bmatrix}. \] We will use the determinant theorems to find its determinant.

Step 2: Use Determinant Formula for 3x3 Matrices

For a 3x3 matrix \(\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \), the determinant is computed as: \[a(ei-fh) - b(di-fg) + c(dh-eg).\]

Step 3: Assign Variables from the Matrix

For our given matrix, identify: \[a=4, b=8, c=0,\]\[d=-1, e=-2, f=1,\]\[g=2, h=4, i=3.\]

Step 4: Calculate Each Component

Substitute the matrix elements into the determinant formula: For \(a(ei-fh)\), calculate: \[4((-2)(3) - 1(4)) = 4(-6-4) = 4(-10) = -40.\]For \(b(di-fg)\), calculate: \[8((-1)(3) - 1(2)) = 8(-3-2) = 8(-5) = -40.\]For \(c(dh-eg)\), calculate: \[0((-1)(4) - (-2)(2)) = 0.\]

Step 5: Compute the Determinant

Add the calculated components together: \[-40 - (-40) + 0 = -40 + 40 + 0 = 0.\] The determinant of the matrix is \(0\).

Key Concepts

Determinant Theorems

Determinant theorems are crucial in matrix algebra, providing methods to calculate the determinant of a matrix, which is a special number that can tell us important information about a matrix. For example, it can reveal whether a matrix is invertible. These theorems simplify the process into manageable steps, essential for more advanced matrix operations.

When dealing with a 3x3 matrix, one particular theorem tells us to use a specific formula. This formula involves multiplying and subtracting the elements in a particular pattern. Understanding this pattern is key:

• The theorem helps provide a systematic approach, particularly important when working with larger and more complex matrices.
• Knowing how to apply these theorems efficiently can save time and reduce errors.
• They pave the way for deeper studies in areas like eigenvectors and eigenvalues.

By mastering determinant theorems, students gain tools for solving a wide range of mathematical problems.

3x3 Matrix

A 3x3 matrix is a grid of numbers arranged into three rows and three columns. It represents a system of linear equations and can encapsulate transformations in three-dimensional space.

In our case, we considered a matrix \[\begin{bmatrix} 4 & 8 & 0 \ -1 & -2 & 1 \ 2 & 4 & 3 \\end{bmatrix}.\]

Each element influences calculations in special ways:

• The elements are labeled as \(a\), \(b\), \(c\), \(d\), \(e\), \(f\), \(g\), \(h\), and \(i\).
• Using these variables, we plug them into the determinant formula to find specific results.
• Understanding the layout and roles of each part is essential for performing deeper matrix operations.

This matrix provides a starting point for exploring more complex concepts in mathematics and physics.

Matrix Algebra

Matrix algebra is a field of mathematics where matrices are used as the main objects of study. It encompasses operations such as addition, subtraction, and importantly, finding determinants.

In matrix algebra, every operation is grounded in important properties and rules:

• Determining if a matrix has an inverse relies on calculating its determinant.
• A determinant of zero, as seen in our example, indicates that the matrix cannot be inverted.
• Matrix multiplication, another key operation, can be heavily influenced by the determinant.

By learning matrix algebra, you can solve real-world problems, ranging from computer graphics transformations to statistical data analysis, making it a vital tool in both academic and professional settings.