Problem 37 - Intermediate Algebra

Question

Evaluate each determinant. $$ \left|\begin{array}{lll} 1 & 2 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 1.

Step 1: Understanding the Structure

The given matrix is a triangular matrix, specifically an upper triangular matrix because all entries below the main diagonal are zero.

Step 2: Identifying Triangular Determinant Property

For an upper or lower triangular matrix, the determinant is the product of its diagonal elements.

Step 3: Calculating the Determinant

Multiply the diagonal elements of the matrix. The elements on the diagonal are 1, 1, and 1. Thus, the determinant is calculated as follows: \( 1 \times 1 \times 1 = 1 \).

Key Concepts

Triangular Matrix

A triangular matrix is a special type of square matrix. It is named for its shape, as it essentially forms a triangle with zeros filling a specific part of the matrix. Understanding the basics of triangular matrices can help simplify calculations, such as finding the determinant. In a general sense, there are two main types: lower triangular and upper triangular matrices. Let’s explore them:

• **Lower Triangular Matrix**: The elements above the main diagonal are all zeros. Only the diagonal and the elements below it can be non-zero.
• **Upper Triangular Matrix**: The focus of our exercise, where all elements below the main diagonal are zeros. This type is what we are dealing with in the original problem.

Notice that triangular matrices, whether upper or lower, have zeros arranged in a way that when drawn out, resemble a triangle. This form is crucial, because it allows for easier calculations of determinants, especially with larger matrices.

Upper Triangular Matrix

An upper triangular matrix is a square matrix in which all the elements below the main diagonal are zero. The special structure of this matrix type simplifies many mathematical operations, notably determinant calculation. In an upper triangular matrix, the diagonal is composed of the important elements, as they are key in determinant evaluation.

Here are some key points about upper triangular matrices:

• The matrix's determinant equals the product of the diagonal elements. This means that instead of complex computations, you're simply multiplying together the diagonal numbers.
• If any diagonal element is zero, the determinant will also be zero. This provides a quick insight into matrix properties without extensive calculations.

Knowing these properties can be particularly helpful in linear algebra, where such matrices often appear in system solutions, eigenvalues, and more. Recognizing and using upper triangular matrices strategically can simplify both calculations and problem-solving processes.

Diagonal Elements

In the context of square matrices, diagonal elements play a pivotal role in many computations. These elements are located on the main diagonal, which stretches from the top-left to the bottom-right of the matrix. Let's explore why they are so important:

• **Role in Determinant Calculation**: In upper (and lower) triangular matrices, like the one in our original exercise, the determinant is calculated by multiplying all the diagonal elements. This property makes calculation significantly quicker and simpler than other methods.
• **Impact on Properties**: The values in the diagonal can influence matrix properties, such as invertibility. For example, if a matrix has zeros on its diagonal, this typically makes the matrix non-invertible (singular).

Understanding the impact and uses of diagonal elements can greatly enhance your ability to work with and interpret matrices efficiently. In many mathematical and applied contexts, diagonal elements serve as indicators for properties and solutions, streamlining processes in linear algebra and matrix theory.