Problem 36 - Intermediate Algebra

Question

Evaluate each determinant. $$ \left|\begin{array}{lll} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the matrix is -48.

Step 1: Understanding the Determinant Formula

For a 3x3 matrix, the determinant can be calculated using the formula:\[\text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \]where the matrix is:\[\begin{pmatrix} a & b & c \d & e & f \g & h & i \end{pmatrix}\]

Step 2: Identifying Values

Identify the values from the given matrix to use in the formula:\[a = 1, \ b = 4, \ c = 7 \ d = 2, \ e = 5, \ f = 8 \ g = 3, \ h = 6, \ i = 9\]Substitute these values into the determinant formula.

Step 3: Calculate Each Term Separately

Substitute the values into the determinant formula equation and calculate each part:\[1(5 \cdot 9 - 6 \cdot 8) = 1(45 - 48) = 1(-3) = -3\]\[4(2 \cdot 9 - 3 \cdot 8) = 4(18 - 24) = 4(-6) = -24\]\[7(2 \cdot 6 - 3 \cdot 5) = 7(12 - 15) = 7(-3) = -21\]

Step 4: Combine Results

Sum the results of each part according to the formula:\[-3 - 24 - 21 = -48\]Thus, the determinant of the matrix is \(-48\).

Key Concepts

Understanding the 3x3 Matrix

A 3x3 matrix is a square array of numbers with three rows and three columns. Each element in the matrix can be identified by its position, usually described using the row number followed by the column number. In our case, the matrix looks like this:\[\begin{pmatrix} 1 & 4 & 7 \2 & 5 & 8 \3 & 6 & 9 \end{pmatrix}\]The numbers inside the matrix are referred to as elements. These elements can be variables or specific numbers, and they are used to perform various mathematical operations, including finding the determinant. Studying 3x3 matrices offers a foundational understanding for more complex matrix operations used in higher-level mathematics and various scientific computations.

Exploring the Determinant Formula for 3x3 Matrices

The determinant is a special number that can be calculated from a square matrix. For a 3x3 matrix, the formula to calculate the determinant is:\[det(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\]In this formula, each letter stands for an element from the matrix. The letters \(a, b, c, d, e, f, g, h,\) and \(i\) correspond to specific elements within the 3x3 matrix. Calculating the determinant involves finding the products of the elements in specific patterns and then summing and subtracting these products.

• \(a(ei-fh)\): This term focuses on the top left corner element \(a\) and combines it with a pattern involving \(e, i, f,\) and \(h\).
• \(- b(di-fg)\): Here, the middle element \(b\) is used to form a different pattern.
• \(+ c(dh-eg)\): Finally, \(c\) is used in another unique pattern.

This formula is essential, as it allows us to calculate the determinant correctly and determine properties of the matrix, such as whether it is invertible.

Mastering Matrix Evaluation using Determinants

Matrix evaluation, particularly using determinants, involves substituting values into the determinant formula and performing the calculations step-by-step. Let's break down the evaluation process:1. **Identify Values:** Pull the elements from your matrix that correspond to \(a, b, c,\) etc. For our example, \[a = 1, \ b = 4, \ c = 7, \ d = 2, \ e = 5, \ f = 8, \ g = 3, \ h = 6, \ i = 9\]2. **Substitute and Calculate:** Place these values into the determinant formula and solve each term:

• \(1(5 \, \cdot \, 9 - 6 \, \cdot \, 8) = -3\)
• \(4(2 \, \cdot \, 9 - 3 \, \cdot \, 8) = -24\)
• \(7(2 \, \cdot \, 6 - 3 \, \cdot \, 5) = -21\)

3. **Combine Results:** Add together the calculated values:\[-3 - 24 - 21 = -48\]By following this structured approach, you can systematically evaluate the determinant of a 3x3 matrix. This process is useful in various practical and theoretical applications, such as solving equations and analyzing system stability.