Problem 13 - A Graphical Approach to College Algebra

Question

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}4 & -7 & 8 \\\2 & 1 & 3 \\\\-6 & 3 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 186.

Step 1: Understand the determinant formula

The determinant of a 3x3 matrix \( A \), denoted as \( \text{det}(A) \), is calculated using the formula: \[ \text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg) \]where the matrix is \[ A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \] and \( a, b, c, d, e, f, g, h, i \) are the elements of the matrix.

Step 2: Identify matrix elements

For the matrix \[ \begin{bmatrix} 4 & -7 & 8 \ 2 & 1 & 3 \ -6 & 3 & 0 \end{bmatrix} \],assign the elements as follows:\( a = 4, b = -7, c = 8, d = 2, e = 1, f = 3, g = -6, h = 3, \) and \( i = 0 \).

Step 3: Apply the determinant formula

Substitute the values into the determinant formula:\[\text{det}(A) = 4(1 imes 0 - 3 imes 3) - (-7)(2 imes 0 - 3 imes -6) + 8(2 imes 3 - 1 imes -6)\]Simplify each term step-by-step.

Step 4: Simplify each term individually

Calculate each term in the expression: 1. Calculate \( 4(1 imes 0 - 3 \times 3) = 4(0 - 9) = 4(-9) = -36 \).2. Calculate \( -(-7)(2 imes 0 - 3 imes -6) = 7(0 + 18) = 7 \times 18 = 126 \).3. Calculate \( 8(2 \times 3 - 1 \times -6) = 8(6 + 6) = 8 \times 12 = 96 \).

Step 5: Combine the results of the terms

Add all the results together to find the determinant:\(\text{det}(A) = -36 + 126 + 96 = 186\).

Key Concepts

3x3 Matrix

Understanding a 3x3 matrix is crucial when learning about determinants and matrix algebra. A 3x3 matrix consists of three rows and three columns of numbers. This is often seen as a square matrix, meaning the number of rows is equal to the number of columns.

The 3x3 matrix can be represented as follows:

• The first row contains the elements: \( a, b, c \)
• The second row contains the elements: \( d, e, f \)
• The third row contains the elements: \( g, h, i \)

These elements are crucial in calculating the determinant, which provides important information about the matrix, such as whether it is invertible. A non-zero determinant indicates an invertible matrix, while a zero determinant reflects a singular (non-invertible) matrix.

Visualizing or drawing the matrix structure can help understand how each element's position contributes to the determinant's calculation, especially when using the formula.

Matrix Algebra

Matrix algebra involves operations that can be performed on matrices, such as addition, subtraction, multiplication, and finding determinants. Calculating the determinant is a fundamental concept in matrix algebra, as it has implications in solving systems of linear equations and understanding matrix properties.

Determinants are calculated using specific formulas that depend on the size of the matrix. For a 3x3 matrix, the formula incorporates products and sums of its elements. This method involves evaluating certain products and their respective permutations:

• Calculate the products including \( a(ei − fh) \)
• Followed by \(- b(di − fg) \)
• Lastly, add \(+ c(dh − eg) \)

In matrix algebra, understanding how to properly manipulate these elements is key. This ensures accurate computations and a better grasp of the matrix's characteristics, such as its ability to transform or solve equations.

Elementary Algebra

Elementary algebra is the foundation upon which more advanced mathematical concepts are built, including matrices and determinants. It involves basic mathematical operations such as addition, subtraction, multiplication, and division.

In the context of determinant calculation, elementary algebra handles the arithmetic part of the formula. It ensures that the numbers are correctly calculated step-by-step:

• Add or subtract terms such as \( (ei − fh), (di − fg), (dh − eg) \)
• Multiply these results by \( a, b, c \)
• Finally, sum up these products to obtain the determinant value.

The step-by-step approach in elementary algebra is vital in minimizing errors, especially in larger matrices, where the number of calculations increases. Emphasizing clarity and precision in calculations bridges the gap between understanding simple arithmetic and mastering more complex matrix operations.