Problem 57 - Precalculus with Limits

Question

In Exercises 55-62, use the matrix capabilities of a graphing utility to evaluate the determinant. \(\left| \begin{array}{r} 7 & 0 & -14 \\ -2 & 5 & 4 \\ -6 & 2 & 12 \end{array} \right|\)

Step-by-Step Solution

Answer

The determinant of the given matrix is -84.

Step 1: Identify the matrix

We identify the matrix for which we want to find determinant. It is a 3x3 matrix. The matrix, denoted as A, is \[A = \left| \begin{array}{r} 7 & 0 & -14 \ -2 & 5 & 4 \ -6 & 2 & 12 \end{array} \right|\]

Step 2: Applying the formula

For a 3x3 matrix, the determinant is calculated using the formula: det(A) = aei + bfg + cdh - ceg - bdi - afh. Here the letters represent elements of the matrix. Applying this formula, we can calculate the determinant \[det(A) = 7*5*12 + 0*-2*12 + -14*5*-6 - -14*5*12 - 0*-6*-14 - 7*-2*-6\]

Step 3: Calculate

Perform all the multiplication and addition operations: \[det(A) = 420 + 0 + 420 - 840 - 0 - 84 = -84\]

Key Concepts

3x3 matrix

A 3x3 matrix is a square block of numbers with three rows and three columns. Each position in the matrix is identified by its row and column number, creating a grid of 9 elements in total. These matrices are essential in solving various computational and real-world problems. The arrangement of elements is very important as it determines the operations we can perform later. For example, the matrix given in the problem is:

• 7, 0, -14
• -2, 5, 4
• -6, 2, 12

We denote this entire block as matrix A. Each element in a 3x3 matrix plays a special role when calculating different matrix properties, such as the determinant.

matrix operations

Matrix operations include a wide variety of calculations, such as addition, multiplication, and the finding of determinants or inverses. These operations follow specific rules and often serve as building blocks for more complex calculations in mathematics. In the context of our exercise:

• Finding the determinant involves a special formula that applies only to square matrices. For a 3x3 matrix, this formula takes into account all nine elements in a structured sequence of multiplications and additions/subtractions.

Understanding matrix operations is crucial because each operation can reveal different properties about the matrix and what it represents.

graphing utility

Graphing utilities are tools, often found in graphing calculators or software programs, that assist in performing complex calculations with matrices. They can quickly compute determinants, inverses, and more. These tools are incredibly useful:

• They save time by automating tedious calculations.
• They reduce errors that might occur in manual computations.

For this particular problem, a graphing utility can compute the determinant of the given matrix efficiently, freeing you from manual calculations and reducing possible computational mistakes. Just input matrix A, and the utility calculates \(-84\) right away!

linear algebra

Linear algebra is the branch of mathematics that focuses on vector spaces and matrix theory. It is a fundamental area, underpinning many other mathematical disciplines and practical applications like physics, computer science, and engineering. Understanding linear algebra helps us:

• Analyze systems of linear equations, which are common in modeling real-world scenarios.
• Perform transformations and operations on matrices for various uses

The computation of determinants, like in this exercise, is one task within linear algebra that allows us to determine characteristics such as matrix invertibility and solutions' uniqueness in systems of equations. Studying linear algebra gives insight into how these structured methods can simplify complex problems.