Problem 38 - Precalculus

Question

Evaluate each \(3 \times 3\) determinant. $$\left|\begin{array}{rrr} -7 & 2 & 5 \\ \frac{7}{8} & 3 & 4 \\ -1 & 4 & 6 \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant is 40.

Step 1: Recall the Determinant Formula

For a 3x3 matrix \( A = \left[ \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \right] \), the determinant \( \det(A) \) is calculated as: \( a(ei - fh) - b(di - fg) + c(dh - eg) \).

Step 2: Apply the Formula

Using the formula from Step 1, substitute the values from the matrix:\[ -7(3 \cdot 6 - 4 \cdot 4) - 2\left(\frac{7}{8} \cdot 6 - 4 \cdot 4\right) + 5\left(\frac{7}{8} \cdot 4 - 3 \cdot (-1)\right)\]

Step 3: Calculate Each Term

First calculate each component inside the parentheses:- For \( 3 \cdot 6 - 4 \cdot 4 \), compute it as: \( 18 - 16 = 2 \).- For \( \frac{7}{8} \cdot 6 - 4 \cdot 4 \), compute it as: \( \frac{42}{8} - 16 = \frac{42}{8} - \frac{128}{8} = -\frac{86}{8} \).- For \( \frac{7}{8} \cdot 4 - 3 \cdot (-1) \), compute it as: \( \frac{28}{8} + 3 = \frac{28}{8} + \frac{24}{8} = \frac{52}{8} \).

Step 4: Evaluate the Expression

Substitute the results back into the expression and evaluate:\(-7 \cdot 2 - 2 \cdot \left(-\frac{86}{8}\right) + 5 \cdot \left(\frac{52}{8}\right)\)Calculate each:- \(-7 \cdot 2 = -14\)- \(-2 \cdot -\frac{86}{8} = \frac{172}{8} = 21.5\)- \(5 \cdot \frac{52}{8} = \frac{260}{8} = 32.5\)

Step 5: Sum the Results

Add up the results to find the determinant:\(-14 + 21.5 + 32.5 = 40\)

Key Concepts

Matrix Operations

Matrix operations are fundamental procedures in linear algebra that involve manipulating matrices to solve algebraic problems. One essential matrix operation is finding the determinant, a scalar value that provides information about the matrix. Understanding these operations is crucial for solving systems of linear equations and performing transformations.

Some basic operations include:

• **Addition**: Combining matrices of the same size by adding their corresponding elements.
• **Subtraction**: Similar to addition, but involves subtracting corresponding elements of matrices.
• **Multiplication**: Involves either multiplying a matrix by a scalar or another matrix, requiring that the number of columns in the first matrix equals the number of rows in the second.

While matrix addition and subtraction are straightforward, multiplication is more complex and requires careful attention to the order and dimensions of matrices involved. The determinant, calculated through specific operations, plays a critical role in understanding matrix properties and behavior.

Linear Algebra

Linear algebra is a branch of mathematics that explores vectors, matrices, and linear transformations. It serves as the foundation for many applications in science and engineering, including computer graphics, physics, and statistics.

Key concepts include:

• **Vectors**: Quantities with both magnitude and direction, used in 2D and 3D space.
• **Matrices**: Rectangular arrays of numbers, symbols, or expressions. Used to represent linear transformations and systems of linear equations.
• **Linear Transformations**: Functions mapping vectors to other vectors, usually involving matrix multiplication.

Understanding these elements is essential for expressing systems in a simplified form, making complex calculations more manageable. Linear algebra allows mathematically describing and solving problems related to space and dimensional changes.

Determinants Calculation

Determinant calculation is a crucial concept in linear algebra, often used to determine characteristics of matrices such as invertibility and scaling factors of linear transformations.

For a 3x3 matrix, the determinant is obtained using the formula:\[det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]This involves computing specific combinations of matrix elements to assess the matrix's properties. Let's break down the calculation for clarity:

• Identify each element in the given matrix and assign them to respective placeholders in the formula.
• Calculate each set of terms inside parentheses (e.g., \(ei - fh\)), ensuring careful attention to order and signs.
• Substitute these values back into the determinant expression, performing each multiplication and addition or subtraction.

By following these steps, anyone can determine the scalar value that represents the essence of the matrix, which is pivotal for understanding whether a matrix can be inversed or how certain transformations affect it.