Problem 37 - Precalculus: Mathematics for Calculus

Question

Evaluate the determinant, using row or column operations whenever possible to simplify your work. $$\left|\begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 0 & 2 & 4 & 6 & 8 \\ 0 & 0 & 3 & 6 & 9 \\ 0 & 0 & 0 & 4 & 8 \\ 0 & 0 & 0 & 0 & 5 \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant is 120.

Step 1: Understanding Determinant of a Triangular Matrix

The given matrix is an upper triangular matrix, meaning that all entries below the main diagonal are zero. For such matrices, the determinant is simply the product of the diagonal elements. We will verify this property and calculate the determinant.

Step 2: Calculate the Diagonal Product

For the given matrix, the diagonal elements are 1, 2, 3, 4, and 5. Calculate the product: \[ 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \]

Step 3: Simplifying the Product

Carry out the multiplication step by step: \[ 1 \cdot 2 = 2 \]\[ 2 \cdot 3 = 6 \]\[ 6 \cdot 4 = 24 \]\[ 24 \cdot 5 = 120 \]

Step 4: Writing the Result

The determinant of the matrix is the product of its diagonal elements, which is calculated to be 120.

Key Concepts

Upper Triangular Matrix

An upper triangular matrix is a special type of square matrix where all the elements below the main diagonal are zero. In simpler terms, if you look at the matrix closely, you will only find non-zero numbers on the diagonal and above. This makes certain calculations quite straightforward, especially when finding determinants.

Some key points about upper triangular matrices include:

• The ease of determining certain properties, such as determinants, as seen in this exercise.
• The matrix shape is particularly useful in solving systems of linear equations and other matrix operations.

Recognizing an upper triangular matrix can save you a lot of time during calculations. Since the lower part of the matrix plays no role in determinant calculation, it simplifies the process significantly.

Diagonal Elements

The diagonal elements in a matrix refer to those that lie on the main diagonal — this is the line stretching from the top left to the bottom right corner. These elements play a crucial role when calculating the determinant of an upper triangular matrix.

Why are diagonal elements important?

• They allow for quick computation of the determinant in triangular matrices; you just multiply them together!
• They often represent key values in practical applications, like eigenvalues in linear algebra.

For example, in our exercise, the diagonal elements are 1, 2, 3, 4, and 5. When calculating the determinant of the matrix, these are the only values we need, simplifying the process immensely.

Matrix Operations

Matrix operations encompass a variety of procedures that include addition, subtraction, multiplication, and determinant calculation, among others. When dealing with matrices, understanding these basic operations is crucial for handling more complex mathematical problems.

In our example with an upper triangular matrix, knowing matrix operations helps you:

• Simplify matrix computations, as certain operations become trivial in special types of matrices like upper triangular ones.
• Transform matrices into more manageable forms, utilizing operations like row and column manipulations.

Matrix operations form the backbone of many mathematical and engineering calculations, making them a fundamental skill in numerous fields. As seen in the exercise, knowing just which operations to apply can make the determinant computation straightforward and efficient.