Problem 60 - College Algebra and Calculus: An Applied Approach

Question

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$ \left|\begin{array}{rrrrr} -2 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & -4 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the given matrix is 48

Step 1: Identify the Matrix Type

The given Matrix is a diagonal matrix. This is clear since all elements in the matrix are equal to zero, except the diagonal elements.

Step 2: Calculate the Determinant

The determinant of a diagonal matrix is simply the product of its diagonal elements. Hence, calculate the determinant of the matrix by multiplying all the diagonal elements. So, Determinant \( = (-2) * 3 * (-1)* 2 * (-4)

Step 3: Simplify the Product

Now, simplify the product calculated in Step 2. Determinant \(= 48\)

Key Concepts

Diagonal Matrix

A diagonal matrix is a special type of square matrix where all the elements outside the main diagonal are zero. Mathematically, this means that for a diagonal matrix \( A \), all \( a_{ij} = 0 \) whenever \( i eq j \). Here are a few important features of diagonal matrices:

• Simple structure that reveals properties easily.
• Determinant is easy to calculate, just multiply the elements on the diagonal.
• Easy for computations, often used in simplifying problems.

Diagonal matrices are significant in linear algebra due to their simplicity, making many matrix operations more streamlined.

Matrix Multiplication

Matrix multiplication is a process for combining two matrices to produce another matrix. To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Here are the steps:

• Identify Dimensions: Make sure dimensions are compatible for multiplication.
• Calculate Entry: Each entry in the resulting matrix is the dot product of a row from the first matrix and a column from the second matrix.
• Repeat Process: Continue for all rows and columns.
• Resulting Matrix: The new matrix has dimensions of the rows from the first matrix and columns from the second matrix.

Matrix multiplication is a crucial operation in algebra, especially for linear transformations and systems of equations. It can seem tricky at first, but practice makes it easier.

Graphing Utility

A graphing utility is a computational tool, often a software or calculator, designed to solve mathematical problems and visualize calculations efficiently. Graphing utilities can handle complex operations beyond basic arithmetic. Here are some functions you might use:

• Matrix Operations: Calculating determinants, inverses, and other matrix properties.
• Graphing Functions: Plotting functions to visualize their behavior.
• Complex Calculations: Solving algebraic and calculus problems.

Using a graphing utility is essential when you need to calculate the determinant of a large matrix or visualize data that can't be easily plotted by hand. They save time and provide accurate results.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It's fundamental in solving equations and modeling real-world phenomena. Some key aspects include:

• Expressions and Equations: Formulas like \( x + 3 = 7 \) to express relationships.
• Variables: Symbols that stand in for unknown values.
• Operations: Addition, subtraction, multiplication, and division applied to numbers and variables.
• Patterns and Functions: Investigating how changing one quantity affects another.

Algebra provides the tools to model situations, analyze relationships, and solve equations, making it a pillar of modern mathematics.